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

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{y^2}{x^3}\iff\dfrac{\mathrm dy}{y^2}=\dfrac{\mathrm dx}{x^3}$

Integrating both sides gives

$-\dfrac1y=-\dfrac1{2x^2}+C$

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

$-\dfrac11=-\dfrac1{2(1)^2}+C\implies C=-\dfrac12$

so that the solution is

$-\dfrac1y=-\dfrac1{2x^2}-\dfrac12\implies y=\dfrac1{\frac1{2x^2}+\frac12}=\dfrac{2x^2}{1+x^2}$