Question

Find the specific solution of the differential equation dy/dx= 4y/x^2 with condition y(-4)=1
A. y=-1-4/x
B. y=-e^1/x
C.y=e^(-4/x)
D. None of these

Answer 1

This ODE is separable:

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{4y}{x^2}\implies\dfrac{\mathrm dy}y=\dfrac4{x^2}\,\mathrm dx$

Integrating both sides gives

$\ln|y|=-\dfrac4x+C$

Given the initial condition $y(-4)=1$ we find

$\ln|1|=-\dfrac4{-4}+C\implies C=-1$

so that the particular solution is

$\ln|y|=-\dfrac4x-1$

$\implies y=e^{-(1+4/x)})$

so the answer is D.