Question

Obtain the general solution to the equation. (x^2+10) + xy = 4x=0 The general solution is y(x) = ignoring lost solutions, if any.

Answer 1

`y(x)=4+(C)/(√(x^2+10))`

Explanation:

We are given that a differential equation

`(x^2+10)y'+xy-4x=0`

We have to find the general solution of given differential equation

`y'+(x)/(x^2+10)y-(4x)/(x^2+10)=0`

`y'+(x)/(x^2+10)y=4(x)/(x^2+10)`

Compare with

`y'+P(x) y=Q(x)`

We get

`P(x)=(x)/(x^2+10)`

`Q(x)=(4x)/(x^2+10)`

I.F=
`e^{\int(x)/(x^2+10) dx}=e^{(1)/(2)ln(x^2+10)}`

`e^(ln\sqrt(x^2+10))=√(x^2+10)`

`y\cdot √(x^2+10)=\int (4x)/(x^2+10)* √(x^2+10) dx+C`

`y\cdot √(x^2+10)=\int (4x)/(√(x^2+10))+C`

`y\cdot √(x^2+10)=4√(x^2+10)+C`

`y(x)=4+(C)/(√(x^2+10))`