I'm trying to solve a second order ODE for [tex]y(x)[/tex] to show that the solution is [tex]y(x)=sin(x)/x[/tex]. We've been told to use the substitution [tex]y(x)=h(x)/x[/tex]. I've got to the stage of solving for [tex]h(x)[/tex], arriving at [tex]h''(x)=-x[/tex]. Using the general solution, [tex]h(x)=A sin(x) + B cos(x)[/tex] and substiting this into the original equation for [tex]y(x)[/tex] I get [tex]y(x)=(A sin(x) + B cos(x))/x[/tex]
So all that's left to do it seems is to use the inital conditions to show A=1 and B=0 however the problem says to use the inital condition [tex]y(0)=1[/tex]. This doesn't make sense to me since y is singular at x=0. Is this inital condition a mistake or am I missing something?
The Attempt at a Solution