Question

1.Solve the differential equation dy/dx= y^2/x^3 for y=f(x) with the condition y(1) = 1.

Answer 1

This ODE is separable, so you can write


`(\mathrm dy)/(\mathrm dx)=(y^2)/(x^3)\iff(\mathrm dy)/(y^2)=(\mathrm dx)/(x^3)`

Integrating both sides gives


`-\frac1y=-\frac1{2x^2}+C`

and given the initial condition
`y(1)=1`, you have


`-\frac11=-\frac1{2(1)^2}+C\implies C=-\frac12`

so that the solution is


`-\frac1y=-\frac1{2x^2}-\frac12\implies y=\frac1{\frac1{2x^2}+\frac12}=(2x^2)/(1+x^2)`