Question

I have
`y = x^(2) + (c)/( x^(2) )` and for any value of c, a solution of the equation is [tex]xy' + 2y = 4 x^{2} , (x\ \textgreater \ 0)

[/tex] and I am supposed to find the value of c for which y(3)=1. What I did was I took the differential of the first equation to get y' and substituted it in the second equation, along with the value for x (which is 3), to solve for c but when I put the value I got for c and the value of x into the first equation, I don't get y = 1. Could anyone please help me see what I'm doing wrong?

Answer 1

Given
`y(3)=1`, the solution gives the equation


`1=3^2+\frac c{3^2}\implies1=9+\frac c9\implies c=-72`

so that the particular solution is


`y=x^2-(72)/(x^2)`

To verify that this solution is correct, differentiate it, then plug it and its derivative into the ODE and arrive at an identity.


`y'=2x+(144)/(x^3)`

`\implies xy'=2x^2+(144)/(x^2)`


`xy'+2y=4x^2\iff \left(2x^2+(144)/(x^2)\right)+2\left(x^2-(72)/(x^2)\right)=4x^2\iff 4x^2=4x^2`

which is true for all
`x>0`.