@pagan Hi pagan, thank you for your answer and patience. I tried your code and it worked for the specific case. By the way, when I changed the code to "assuming y(0)::finite", Maple aslo worked out the result. However, I have a follow-up question on the same subject. Please refer to the following lines of code.

restart;

dsolve({diff(y(x),x,x)+diff(y(x),x)/x-y(x)*(n^2)/(x^2)=-4*(n^2)/(x^2),y(1)=0,y(0)=y(0)})
    assuming y(0)::finite;

Maple returned "y(x)=x^n*C1+x^(-n)(-C1-4)+4". My intention was to ask Maple to solve the ODE assuming that n was a positive integer so that C1=-4 and the coefficient in front of x^(-n) would disappear. I tried to juxtapose another condition "n::posint" along with y(0)::finite in my assumption argument, but Maple complained that no assumption can be made about n as it is a constant.

Have you any suggestions? Thank you again for your help.

@pagan Hello, thank you all for your replies. However, I have not yet managed to obtain a result with the following codes:

restart;
assume(K<infinity);
dsolve({diff(y(x),x,x)+(diff(y(x),x))/x+8,y(1)=0,y(0)=K});

As you can see, I wish to specify the condition y(0) is finite when I execute dsolve. Any suggestions?

@pagan Hello, thank you all for your replies. However, I have not yet managed to obtain a result with the following codes:

restart;
assume(K<infinity);
dsolve({diff(y(x),x,x)+(diff(y(x),x))/x+8,y(1)=0,y(0)=K});

As you can see, I wish to specify the condition y(0) is finite when I execute dsolve. Any suggestions?

I understand that x in ln(x) can only be a positive number. However, that does not resolve my original question. The function y(x)=-2*x^2+C1*ln(x) is a solution to a physical problem where x=0 exists in the physical domain. As a result, for the solution to be physically sensible, C1 must be zero, therefore eliminating the ln(x) term. My question is how to set up the problem so that Maple knows y(x) is finite at x=0 and therefore assigns a value of 0 to C1 when solving the ODE.

I understand that x in ln(x) can only be a positive number. However, that does not resolve my original question. The function y(x)=-2*x^2+C1*ln(x) is a solution to a physical problem where x=0 exists in the physical domain. As a result, for the solution to be physically sensible, C1 must be zero, therefore eliminating the ln(x) term. My question is how to set up the problem so that Maple knows y(x) is finite at x=0 and therefore assigns a value of 0 to C1 when solving the ODE.