## A strange result from dsolve...

Hello guys,

I was just playing around with differential equations, when I noticed that symbolic solution is  different from the numerical.What is the reason for this strange behavior?

ODE := (diff(y(x), x))*(ln(y(x))+x) = 1

sol := dsolve({ODE, y(1) = 1}, y(x))

a := plot(op(2, sol), x = .75 .. 2, color = "Red");
sol2 := dsolve([ODE, y(1) = 1], numeric, range = .75 .. 2);

with(plots);
b := odeplot(sol2, .75 .. 2, thickness = 4);
display({a, b});

Mariusz Iwaniuk

## Solving with HPM...

Hi everyone. I'm going to solve a problem of an article with hpm. well I wrote some initial codes(I uploaded both codes and article). but now I face with a problem. I cant reach to the correct plot that is in the article. could you please help me???

(dont think I am lazy ;))) I found f and g (by make a system with A1 and B1 and solve it i found f[0] and g[0], with p^3 coefficient in A-->f[1] and then with B2 I foud g[1]) and their plot was correct. but the problem is theta and phi and their plots :(( )

Project.mw

2.pdf   this is article

 > restart;
 > lambda:=0.5;K[r]:=0.5;Sc:=0.5;Nb:=0.1;Nt:=0.1;Pr:=10;
 (1)
 > equ1:=diff(f(eta),eta\$4)-R*(diff(f(eta),eta)*diff(f(eta),eta\$2)-f(eta)*diff(f(eta),eta\$2))-2*K[r]*diff(g(eta),eta)=0; equ2:=diff(g(eta),eta\$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta)=0; equ3:=diff(theta(eta),eta\$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2=0; equ4:=diff(phi(eta),eta\$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta\$2)*(Nt/Nb)=0;
 (2)
 > ics:= f(0)=0,D(f)(0)=1,g(0)=0,theta(0)=1,phi(0)=1; f(1)=lambda,D(f)(1)=0,g(1)=0,theta(1)=0,phi(1)=0;
 (3)
 > hpm1:=(1-p)*(diff(f(eta),eta\$4)-2*K[r]*diff(g(eta),eta))+p*(diff(f(eta),eta\$4)-R*(diff(f(eta),eta)*diff(f(eta),eta\$2)-f(eta)*diff(f(eta),eta\$2))-2*K[r]*diff(g(eta),eta))=0; hpm2:=(1-p)*(diff(g(eta),eta\$2)+2*K[r]*diff(f(eta),eta))+p*(diff(g(eta),eta\$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta))=0; hpm3:=(1-p)*(diff(theta(eta),eta\$2))+p*(diff(theta(eta),eta\$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2)=0; hpm4:=(1-p)*(diff(phi(eta),eta\$2)+diff(theta(eta),eta\$2)*(Nt/Nb))+p*(diff(phi(eta),eta\$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta\$2)*(Nt/Nb))=0;
 (4)
 > f(eta)=sum(f[i](eta)*p^i,i=0..1);
 (5)
 > g(eta)=sum(g[i](eta)*p^i,i=0..1);
 (6)
 > theta(eta)=sum(theta[i](eta)*p^i,i=0..1);
 (7)
 > phi(eta)=sum(phi[i](eta)*p^i,i=0..1);
 (8)
 > A:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm1)),p);
 (9)
 > A1:=diff(f[0](eta),eta\$4)-2*K[r]*(diff(g[0](eta),eta))=0; A2:=diff(f[1](eta),eta\$4)-2*K[r]*(diff(g[1](eta),eta))-R*(diff(f[0](eta),eta))*(diff(f[0](eta),eta\$2))+R*f[0](eta)*(diff(f[0](eta),eta\$2))=0;
 (10)
 > icsA1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0; icsA2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;
 (11)
 > B:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm2)),p);
 (12)
 > B1:=diff(g[0](eta),eta\$2)+2*K[r]*(diff(f[0](eta),eta))=0; B2:=diff(g[1](eta),eta\$2)+2*K[r]*(diff(f[1](eta),eta))-R*(diff(f[0](eta),eta))*g[0](eta)+R*f[0](eta)*(diff(g[0](eta),eta))=0;
 (13)
 > icsB1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0; icsB2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;
 (14)
 > C:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm3)),p);
 (15)
 > C1:=diff(theta[0](eta),eta\$2)=0; C2:=diff(theta[1](eta), eta, eta)+Pr*R*f[0](eta)*(diff(theta[0](eta), eta))+Nb*(diff(phi[0](eta), eta))*(diff(theta[0](eta), eta))+Nt*(diff(theta[0](eta), eta))^2=0;
 (16)
 > icsC1:=theta[0](0)=1,theta[0](1)=0; icsC2:=f[0](0)=0,D(f[0])(0)=1,f[1](1)=0,D(f[1])(1)=0,theta[1](0)=0,theta[1](1)=0,phi[0](0)=0,phi[0](1)=0;
 (17)
 > E:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm4)),p);
 (18)
 > E1:=diff(phi[0](eta),eta\$2)+Nt*(diff(theta[0](eta),eta\$2))/Nb=0; E2:=diff(phi[1](eta),eta\$2)+Nt*(diff(theta[1](eta),eta\$2))/Nb+R*Sc*f[0](eta)*(diff(phi[0](eta),eta))=0;
 (19)
 > icsE1:=phi[0](0)=1,phi[0](1)=0; icsE2:=f[0](0)=0,D(f[0])(0)=1,f[1](1)=0,D(f[1])(1)=0,theta[1](0)=0,theta[1](1)=0,phi[1](0)=0,phi[1](1)=0;
 (20)
 >

Project.mw

thanks for your favorits

## Problem with ODE system...

Hi, i am trying to solve my PDEs with HPM method ,but i get strange errors.

first one is :Error, (in trig/reduce/reduce) Maple was unable to allocate enough memory to complete this computation.  Please see ?alloc,

but when i run my last function again,the error chages,let me show you.

restart;
lambda:=0.5;K[r]:=0.5;Sc:=0.5;Nb:=0.1;Nt:=0.1;Pr:=10;
0.5
0.5
0.5
0.1
0.1
10
> EQUATIONS;

equ1:=diff(f(eta),eta\$4)-R*(diff(f(eta),eta)*diff(f(eta),eta\$2)-f(eta)*diff(f(eta),eta\$2))-2*K[r]*diff(g(eta),eta)=0;

equ2:=diff(g(eta),eta\$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta)=0;

equ3:=diff(theta(eta),eta\$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2=0;

equ4:=diff(phi(eta),eta\$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta\$2)*(Nt/Nb)=0;
/  d   /  d   /  d   /  d         \\\\     //  d         \ /  d
|----- |----- |----- |----- f(eta)|||| - R ||----- f(eta)| |-----
\ deta \ deta \ deta \ deta       ////     \\ deta       / \ deta

/  d         \\          /  d   /  d         \\\
|----- f(eta)|| - f(eta) |----- |----- f(eta)|||
\ deta       //          \ deta \ deta       ///

/  d         \
- 1.0 |----- g(eta)| = 0
\ deta       /
/  d   /  d         \\
|----- |----- g(eta)||
\ deta \ deta       //

//  d         \                 /  d         \\
- R ||----- f(eta)| g(eta) - f(eta) |----- g(eta)||
\\ deta       /                 \ deta       //

/  d         \
+ 1.0 |----- f(eta)| = 0
\ deta       /
/  d   /  d             \\               /  d             \
|----- |----- theta(eta)|| + 10 R f(eta) |----- theta(eta)|
\ deta \ deta           //               \ deta           /

/  d           \ /  d             \
+ 0.1 |----- phi(eta)| |----- theta(eta)|
\ deta         / \ deta           /

2
/  d             \
+ 0.1 |----- theta(eta)|  = 0
\ deta           /
/  d   /  d           \\                /  d           \
|----- |----- phi(eta)|| + 0.5 R f(eta) |----- phi(eta)|
\ deta \ deta         //                \ deta         /

/  d   /  d             \\
+ 1.000000000 |----- |----- theta(eta)|| = 0
\ deta \ deta           //
> BOUNDARY*CONDITIONS;

ics:=
f(0)=0,D(f)(0)=1,g(0)=0,theta(0)=1,phi(0)=1;
f(1)=lambda,D(f)(1)=0,g(1)=0,theta(1)=0,phi(1)=0;
f(0) = 0, D(f)(0) = 1, g(0) = 0, theta(0) = 1, phi(0) = 1
f(1) = 0.5, D(f)(1) = 0, g(1) = 0, theta(1) = 0, phi(1) = 0
> HPMs;

hpm1:=(1-p)*(diff(f(eta),eta\$4)-2*K[r]*diff(g(eta),eta))+p*(diff(f(eta),eta\$4)-R*(diff(f(eta),eta)*diff(f(eta),eta\$2)-f(eta)*diff(f(eta),eta\$2))-2*K[r]*diff(g(eta),eta))=0;

hpm2:=(1-p)*(diff(g(eta),eta\$2)+2*K[r]*diff(f(eta),eta))+p*(diff(g(eta),eta\$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta))=0;

hpm3:=(1-p)*(diff(theta(eta),eta\$2))+p*(diff(theta(eta),eta\$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2)=0;

hpm4:=(1-p)*(diff(phi(eta),eta\$2)+diff(theta(eta),eta\$2)*(Nt/Nb))+p*(diff(phi(eta),eta\$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta\$2)*(Nt/Nb))=0;

//  d   /  d   /  d   /  d         \\\\
(1 - p) ||----- |----- |----- |----- f(eta)||||
\\ deta \ deta \ deta \ deta       ////

/  d         \\     //  d   /  d   /  d   /  d         \
- 1.0 |----- g(eta)|| + p ||----- |----- |----- |----- f(eta)|
\ deta       //     \\ deta \ deta \ deta \ deta       /

\\\     //  d         \ /  d   /  d         \\
||| - R ||----- f(eta)| |----- |----- f(eta)||
///     \\ deta       / \ deta \ deta       //

/  d   /  d         \\\       /  d         \\
- f(eta) |----- |----- f(eta)||| - 1.0 |----- g(eta)|| = 0
\ deta \ deta       ///       \ deta       //
//  d   /  d         \\       /  d         \\     //  d
(1 - p) ||----- |----- g(eta)|| + 1.0 |----- f(eta)|| + p ||-----
\\ deta \ deta       //       \ deta       //     \\ deta

/  d         \\
|----- g(eta)||
\ deta       //

//  d         \                 /  d         \\
- R ||----- f(eta)| g(eta) - f(eta) |----- g(eta)||
\\ deta       /                 \ deta       //

/  d         \\
+ 1.0 |----- f(eta)|| = 0
\ deta       //
/
/  d   /  d             \\     |/  d   /  d             \
(1 - p) |----- |----- theta(eta)|| + p ||----- |----- theta(eta)|
\ deta \ deta           //     \\ deta \ deta           /

\               /  d             \
| + 10 R f(eta) |----- theta(eta)|
/               \ deta           /

/  d           \ /  d             \
+ 0.1 |----- phi(eta)| |----- theta(eta)|
\ deta         / \ deta           /

2\
/  d             \ |
+ 0.1 |----- theta(eta)| | = 0
\ deta           / /
//  d   /  d           \\
(1 - p) ||----- |----- phi(eta)||
\\ deta \ deta         //

/  d   /  d             \\\     //  d   /  d
+ 1.000000000 |----- |----- theta(eta)||| + p ||----- |-----
\ deta \ deta           ///     \\ deta \ deta

\\                /  d           \
phi(eta)|| + 0.5 R f(eta) |----- phi(eta)|
//                \ deta         /

/  d   /  d             \\\
+ 1.000000000 |----- |----- theta(eta)||| = 0
\ deta \ deta           ///
f(eta)=sum(f[i](eta)*p^i,i=0..1);
f(eta) = f[0](eta) + f[1](eta) p
g(eta)=sum(g[i](eta)*p^i,i=0..1);
g(eta) = g[0](eta) + g[1](eta) p
theta(eta)=sum(theta[i](eta)*p^i,i=0..1);
theta(eta) = theta[0](eta) + theta[1](eta) p
phi(eta)=sum(phi[i](eta)*p^i,i=0..1);
phi(eta) = phi[0](eta) + phi[1](eta) p
> FORequ1;

A:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm1)),p);
/      /  d            \ /  d   /  d            \\
|-1. R |----- f[1](eta)| |----- |----- f[1](eta)||
\      \ deta          / \ deta \ deta          //

/  d   /  d            \\\  3   /
+ R f[1](eta) |----- |----- f[1](eta)||| p  + |
\ deta \ deta          ///      \
/  d            \ /  d   /  d            \\
-1. R |----- f[0](eta)| |----- |----- f[1](eta)||
\ deta          / \ deta \ deta          //

/  d            \ /  d   /  d            \\
- 1. R |----- f[1](eta)| |----- |----- f[0](eta)||
\ deta          / \ deta \ deta          //

/  d   /  d            \\
+ R f[0](eta) |----- |----- f[1](eta)||
\ deta \ deta          //

/  d   /  d            \\\  2   //  d   /  d   /
+ R f[1](eta) |----- |----- f[0](eta)||| p  + ||----- |----- |
\ deta \ deta          ///      \\ deta \ deta \

d   /  d            \\\\       /  d            \
----- |----- f[1](eta)|||| - 1.0 |----- g[1](eta)|
deta \ deta          ////       \ deta          /

/  d            \ /  d   /  d            \\
- 1. R |----- f[0](eta)| |----- |----- f[0](eta)||
\ deta          / \ deta \ deta          //

/  d   /  d            \\\
+ R f[0](eta) |----- |----- f[0](eta)||| p
\ deta \ deta          ///

/  d   /  d   /  d   /  d            \\\\
+ |----- |----- |----- |----- f[0](eta)||||
\ deta \ deta \ deta \ deta          ////

/  d            \
- 1.0 |----- g[0](eta)| = 0
\ deta          /
A1:=diff(f[0](eta),eta\$4)-2*K[r]*(diff(g[0](eta),eta))=0;
A2:=diff(f[1](eta),eta\$4)-2*K[r]*(diff(g[1](eta),eta))-R*(diff(f[0](eta),eta))*(diff(f[0](eta),eta\$2))+R*f[0](eta)*(diff(f[0](eta),eta\$2))=0;
/  d   /  d   /  d   /  d            \\\\       /  d            \
|----- |----- |----- |----- f[0](eta)|||| - 1.0 |----- g[0](eta)| =
\ deta \ deta \ deta \ deta          ////       \ deta          /

0
/  d   /  d   /  d   /  d            \\\\       /  d            \
|----- |----- |----- |----- f[1](eta)|||| - 1.0 |----- g[1](eta)|
\ deta \ deta \ deta \ deta          ////       \ deta          /

/  d            \ /  d   /  d            \\
- R |----- f[0](eta)| |----- |----- f[0](eta)||
\ deta          / \ deta \ deta          //

/  d   /  d            \\
+ R f[0](eta) |----- |----- f[0](eta)|| = 0
\ deta \ deta          //
icsA1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0;
icsA2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;
f[0](0) = 0, D(f[0])(0) = 1, g[0](0) = 0, f[0](1) = 0.5,

D(f[0])(1) = 0, g[0](1) = 0
f[1](0) = 0, D(f[1])(0) = 0, g[1](0) = 0, f[1](1) = 0,

D(f[1])(1) = 0, g[1](1) = 0
>
FORequ2;

B:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm2)),p);
/      /  d            \
|-1. R |----- f[1](eta)| g[1](eta)
\      \ deta          /

/  d            \\  3   /
+ R f[1](eta) |----- g[1](eta)|| p  + |
\ deta          //      \
/  d            \
-1. R |----- f[0](eta)| g[1](eta)
\ deta          /

/  d            \
- 1. R |----- f[1](eta)| g[0](eta)
\ deta          /

/  d            \
+ R f[0](eta) |----- g[1](eta)|
\ deta          /

/  d            \\  2   //  d   /  d
+ R f[1](eta) |----- g[0](eta)|| p  + ||----- |----- g[1](eta)
\ deta          //      \\ deta \ deta

\\       /  d            \        /  d            \
|| + 1.0 |----- f[1](eta)| - 1. R |----- f[0](eta)| g[0](eta)
//       \ deta          /        \ deta          /

/  d            \\     /  d   /  d            \\
+ R f[0](eta) |----- g[0](eta)|| p + |----- |----- g[0](eta)||
\ deta          //     \ deta \ deta          //

/  d            \
+ 1.0 |----- f[0](eta)| = 0
\ deta          /
B1:=diff(g[0](eta),eta\$2)+2*K[r]*(diff(f[0](eta),eta))=0;
B2:=diff(g[1](eta),eta\$2)+2*K[r]*(diff(f[1](eta),eta))-R*(diff(f[0](eta),eta))*g[0](eta)+R*f[0](eta)*(diff(g[0](eta),eta))=0;
/  d   /  d            \\       /  d            \
|----- |----- g[0](eta)|| + 1.0 |----- f[0](eta)| = 0
\ deta \ deta          //       \ deta          /
/  d   /  d            \\       /  d            \
|----- |----- g[1](eta)|| + 1.0 |----- f[1](eta)|
\ deta \ deta          //       \ deta          /

/  d            \
- R |----- f[0](eta)| g[0](eta)
\ deta          /

/  d            \
+ R f[0](eta) |----- g[0](eta)| = 0
\ deta          /
icsB1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0;
icsB2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;
f[0](0) = 0, D(f[0])(0) = 1, g[0](0) = 0, f[0](1) = 0.5,

D(f[0])(1) = 0, g[0](1) = 0
f[1](0) = 0, D(f[1])(0) = 0, g[1](0) = 0, f[1](1) = 0,

D(f[1])(1) = 0, g[1](1) = 0
> FORequ3;

C:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm3)),p);
/
|                /  d                \
|10. R f[1](eta) |----- theta[1](eta)|
\                \ deta              /

/  d              \ /  d                \
+ 0.1 |----- phi[1](eta)| |----- theta[1](eta)|
\ deta            / \ deta              /

2\
/  d                \ |  3   /                /  d
+ 0.1 |----- theta[1](eta)| | p  + |10. R f[0](eta) |-----
\ deta              / /      \                \ deta

\                   /  d                \
theta[1](eta)| + 10. R f[1](eta) |----- theta[0](eta)|
/                   \ deta              /

/  d              \ /  d                \
+ 0.1 |----- phi[0](eta)| |----- theta[1](eta)|
\ deta            / \ deta              /

/  d              \ /  d                \
+ 0.1 |----- phi[1](eta)| |----- theta[0](eta)|
\ deta            / \ deta              /

/
/  d                \ /  d                \\  2   |/
+ 0.2 |----- theta[0](eta)| |----- theta[1](eta)|| p  + ||
\ deta              / \ deta              //      \\

d   /  d                \\
----- |----- theta[1](eta)||
deta \ deta              //

/  d                \
+ 10. R f[0](eta) |----- theta[0](eta)|
\ deta              /

/  d              \ /  d                \
+ 0.1 |----- phi[0](eta)| |----- theta[0](eta)|
\ deta            / \ deta              /

2\
/  d                \ |
+ 0.1 |----- theta[0](eta)| | p
\ deta              / /

/  d   /  d                \\
+ |----- |----- theta[0](eta)|| = 0
\ deta \ deta              //
C1:=diff(theta[0](eta),eta\$2)=0;
C2:=diff(theta[1](eta), eta, eta)+Pr*R*f[0](eta)*(diff(theta[0](eta), eta))+Nb*(diff(phi[0](eta), eta))*(diff(theta[0](eta), eta))+Nt*(diff(theta[0](eta), eta))^2=0;
d   /  d                \
----- |----- theta[0](eta)| = 0
deta \ deta              /
/  d   /  d                \\
|----- |----- theta[1](eta)||
\ deta \ deta              //

/  d                \
+ 10 R f[0](eta) |----- theta[0](eta)|
\ deta              /

/  d              \ /  d                \
+ 0.1 |----- phi[0](eta)| |----- theta[0](eta)|
\ deta            / \ deta              /

2
/  d                \
+ 0.1 |----- theta[0](eta)|  = 0
\ deta              /
icsC1:=theta[0](0)=1,theta[0](1)=0;
icsC2:=theta[1](0)=0,theta[1](1)=0,phi[0](0)=0,phi[0](1)=0;
theta[0](0) = 1, theta[0](1) = 0
theta[1](0) = 0, theta[1](1) = 0, phi[0](0) = 0, phi[0](1) = 0
> FORequ4;

E:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm4)),p);
3 /  d              \   /                /  d
0.5 R f[1](eta) p  |----- phi[1](eta)| + |0.5 R f[0](eta) |-----
\ deta            /   \                \ deta

\                   /  d              \\  2   //
phi[1](eta)| + 0.5 R f[1](eta) |----- phi[0](eta)|| p  + ||
/                   \ deta            //      \\

d   /  d              \\
----- |----- phi[1](eta)||
deta \ deta            //

/  d   /  d                \\
+ 1.000000000 |----- |----- theta[1](eta)||
\ deta \ deta              //

/  d              \\
+ 0.5 R f[0](eta) |----- phi[0](eta)|| p
\ deta            //

/  d   /  d              \\
+ |----- |----- phi[0](eta)||
\ deta \ deta            //

/  d   /  d                \\
+ 1.000000000 |----- |----- theta[0](eta)|| = 0
\ deta \ deta              //
E1:=diff(phi[0](eta),eta\$2)+Nt*(diff(theta[0](eta),eta\$2))/Nb=0;
E2:=diff(phi[1](eta),eta\$2)+Nt*(diff(theta[1](eta),eta\$2))/Nb+R*Sc*f[0](eta)*(diff(phi[0](eta),eta))=0;
/  d   /  d              \\
|----- |----- phi[0](eta)||
\ deta \ deta            //

/  d   /  d                \\
+ 1.000000000 |----- |----- theta[0](eta)|| = 0
\ deta \ deta              //
/  d   /  d              \\
|----- |----- phi[1](eta)||
\ deta \ deta            //

/  d   /  d                \\
+ 1.000000000 |----- |----- theta[1](eta)||
\ deta \ deta              //

/  d              \
+ 0.5 R f[0](eta) |----- phi[0](eta)| = 0
\ deta            /
icsE1:=theta[0](0)=1,theta[0](1)=0,phi[0](0)=1,phi[0](1)=0;
icsE2:=theta[1](0)=0,theta[1](1)=0,phi[1](0)=0,phi[1](1)=0;
theta[0](0) = 1, theta[0](1) = 0, phi[0](0) = 1, phi[0](1) = 0
theta[1](0) = 0, theta[1](1) = 0, phi[1](0) = 0, phi[1](1) = 0

theta[0](eta) = -(152675527/100000000)*eta+1;
152675527
theta[0](eta) = - --------- eta + 1
100000000
U:=f[1](eta)=0;
f[1](eta) = 0
Dsolve(A1,B1,icsA1,icsB1);
Dsolve(A1, B1, icsA1, icsB1)

sys:={ diff(g[0](eta), eta, eta)+1.0*(diff(f[0](eta), eta)) = 0, diff(f[0](eta), eta, eta, eta, eta)-1.0*(diff(g[0](eta), eta)) = 0};
//  d   /  d   /  d   /  d            \\\\
{ |----- |----- |----- |----- f[0](eta)||||
\\ deta \ deta \ deta \ deta          ////

/  d            \
- 1.0 |----- g[0](eta)| = 0,
\ deta          /

/  d   /  d            \\       /  d            \    \
|----- |----- g[0](eta)|| + 1.0 |----- f[0](eta)| = 0 }
\ deta \ deta          //       \ deta          /    /
IC_1:={ f[0](0) = 0, (D(f[0]))(0) = 1, g[0](0) = 0, f[0](1) = .5, (D(f[0]))(1) = 0, g[0](1) = 0,f[0](0) = 0, (D(f[0]))(0) = 1, g[0](0) = 0, f[0](1) = .5, (D(f[0]))(1) = 0, g[0](1) = 0};
{f[0](0) = 0, f[0](1) = 0.5, g[0](0) = 0, g[0](1) = 0,

D(f[0])(0) = 1, D(f[0])(1) = 0}
ans1 := combine(dsolve(sys union IC_1,{f[0](eta),g[0](eta)}),trig);
Error, (in dsolve) expecting an ODE or a set or list of ODEs. Received `union`(IC_1, sys)
>

## Numerical solution of challenging ODE...

Hello everybody.

I'm trying to obtain the numerical solution of a differential equation. Unfortunately, this prove to be quite challenging. I was able to obtain a rough solution using mathematica, but nothing more. The function is strictly increasing (for sure).

Any help is really REALLY appreciated, thanks!

 (1)

 (2)

 (3)

## ODE system in vector form...

I'm working in a tridimensional euclidean space, with vectorial functions of the type:

Fi(t)=<fix(t),fiy(t),fiz(t)>

Fi'(t)=<fix'(t),fiy'(t),fiz'(t)>

The two odes are of the type:

ode1:=K1*F1''(t)=K2*F2(t)&xF3(t)+...

While there are other non-differential vectorial equations like:

eq1:=K4*F4''(t)=(K5*F5(t)&x<0,1,0>)/Norm(F6(t))+..., etc

Is there a way i can input this system in dsolve with vectors instead of scalars? And without splitting everything into its 3 vectorial components? I can't make maple realize some of the Fi(t) functions are vectors, it counts them as scalars and says the number of functions and equations are not the same.

Thank you!

## Replacing variable in ODE...

Hi,

It might be very silly question, but i dont know why it is not working out. So here is the question. In the attached maple shhet when i am trying to substitute eta(t)=epsilon*z(t) then it is not making that susbtitution for differential operator. Apart from that when i m collecting epsilon terms then also it not collecting it.quesiton.mw

Regards

Sunit

## Exact solution of nonlinear ODE? ...

restart:with(plots):
eq:=(diff(f(eta),eta\$2))-a*f(eta)+b*(1+diff(f(eta),eta)^2)^(-1/2)=0;
bc:=f(1)=0,D(f)(0)=0;
ans := dsolve(eq);

## Error when dsolving...

Been working on a diffy q project, new to maple here. Any help is appreciated. Keep getting a similar error.

"Error, (in dsolve/numeric/type_check) insufficient initial/boundary value information for procedure defined problem"

I thought I gave it initial values?

link to screenshot of the error bellow:

http://i.imgur.com/YVE1x7e.jpg

## How to convert ODE system to differential form?...

how to convert system of differential equations to differential form for evalDG?

[a(t)*(diff(c(t), t))+b(t), a(t)*(diff(b(t), t))+c(t)*(diff(b(t), t)), a(t)*(diff(c(t), t))+a(t)*(diff(b(t), t))+b(t)];

when i try eliminate dt which is the denominator

eliminate([a(t)*dc(t) + b(t)*dt,a(t)*db(t)+dt*c(t)*db(t),a(t)*dc(t)+a(t)*db(t)+b(t)*dt],dt);

[{dt = -a(t)/c(t)}, {a(t)*(c(t)*dc(t)-b(t)), a(t)*(db(t)*c(t)+c(t)*dc(t)-b(t))}]

i got two solutions, which one is correct?

a(t)*(c(t)*dc(t)-b(t)), a(t)*(db(t)*c(t)+c(t)*dc(t)-b(t))

does it mean that two have to use together to form a differential form?

update1

with(DifferentialGeometry):
DGsetup([a,b,c], M);
X := evalDG({a*(c*D_c-b), a*(D_b*c+c*D_c-b(t))});
Flow(X,t);
Flow(X, t, ode = true);

got error when run with above result

## Numerical solution of ODE...

Hi

Dear friends

I use the command "dsolve(`union`(deq, initial), numeric, method = lsode)" for solving a fourth order ODE.

But for some numerical values of the parameters the bellow error is occurred:

" an excessive amount of work (greater than mxstep) was done ".

I have three questions:

1- how can I increase the mxstep from default amount (i.e. 500) to a greater value?

2- how can I ensure that the absolute error is less than 10E-6?

3- when I use lsode which way of numerical solution is applied (Euler,midpoint, rk3, rk4, rkf, heun, ... )?

Thanks a lot for your help

## Fourth order differential equation...

Hello evrey one , I need help for solve these equation with boundary conditions

Boundary Conditions

My COde + equation

 >
 >
 >
 >
 (1)
 >
 >
 >
 >
 >

Thank you

## How to rearrange equations...

Hello,

I have a complex set of non linear diff eqns in the form :

y1'' = f(y1',y1,y2'',y2',y2,y3'',y3',y3,y4'',....,y6'',y6',y6,u1,u2,u3,u4) ;

y2'' = f(y1'',y1',y1,y2',y2,y3'',y3',y3,y4'',....,y6'',y6',y6,u1,u2,u3,u4)

and so on ... y6''=(...)

As I want to resolve this coupled systeme in matlab using @ODE45... I wanted the equations in the form : y1''=f(y1',y1,y2',y2,....) and so on ... => X'[] = f(X[],U[])

How can I force maple to rearrange a system of coupled eqns with only the variables i want ?

I know this is possible beacause it is a nonlinear state space model but maple do not work with nonlinear state space model... It give me error when I tried to create statespace model with my non linear diff eqns.

Thanks a lot !

## Solar System Project Code Help...

solarsysem.mw Sorry for the repost but this is my newest document.

I have to create a solar system model on maple by defining a force equation then using the seq function to create a diffeq and then solving those differential equations using the initial conditions with the sun at (0, 0, 0) in xyz coordinates.

It works until my last "ic1" entry and I get an error in dsolve/numeric/process_input

I'm pretty desperate, I'll appreciate any help I can get

## Error when solving ODE system...

hi.please see attached file below and help me.thanksode.mw

## Bounded solution of ODE...

Dear all

I have the following equaion

Eq := diff(phi(x, k), x, x)+(k^2+2*sech(x))*phi(x, k) = 0;

The solution is given by

phi := (I*k-tanh(x))*exp(I*k*x)/(I*k-1);

My question : At what value of k is there a bound state and in this case can we give a simple form of the solution phi(x,k)

With best regards

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