Solution

Kitonum — Wed, 06 Feb 2013 23:44:15 Z

I guess you consider the function of two variables as a scalar field in the plane and the stream lines - the curves in the plane, which at each point has a tangent vector, which coincides with the gradient. If so, then your problem can be solved as follows (psi=3):

psi:=(x,y)->2*y-exp(-x):

C:=3: X:=[seq(i, i=0..1, 0.1)]:

Y:=[seq(solve(psi(X[i], y)=C), i=1..11)]:

V:=[seq(dsolve({diff(x(t), t)=subs(x=x(t), y=y(t), diff(psi(x,y), x)), diff(y(t), t)=subs(x=x(t), y=y(t), diff(psi(x,y), y)), x(0)=X[i], y(0)=Y[i]}), i=1..11)]:

Sol:=evalf(subs(_Z1=0, V));

plot([seq([rhs(Sol[i,1]), rhs(Sol[i,2]), t=0..3], i=1..11)], color=red, thickness=2, labels=[x, y], title="Stream lines for psi =3, x = 0 .. 1", titlefont=[TIMES, ROMAN, 18], view=[0..2, 0..5+C] );

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Solution