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:

`(\mathrm dy)/(\mathrm dx)=(4y)/(x^2)\implies\frac{\mathrm dy}y=\frac4{x^2}\,\mathrm dx`

Integrating both sides gives

`\ln|y|=-\frac4x+C`

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

`\ln|1|=-\frac4{-4}+C\implies C=-1`

so that the particular solution is

`\ln|y|=-\frac4x-1`

`\implies y=e^(-(1+4/x)))`

so the answer is D.