Question

Solve: dy/dx − 2xy = x, with y(0) = 0.

Answer 1

`1 + 2y = x^2`

Explanation:

Given differential equation,

`(dy)/(dx)-2xy=x`

`(dy)/(dx)=x+2xy`

`(dy)/(dx)=x(1+2y)`

`(dy)/(1+2y)=xdx`

Integrating both sides,

`\int (dy)/(1+2y)=\int xdx----(1)`

Put 1 + 2y = u

Differentiating both sides,

2dy = du

`\implies dy=(du)/(2)`

From equation (1),

`(1)/(2) \int (du)/(u)=\int xdx`

`(1)/(2)\log u = \log x + C`

`(1)/(2) \log (1+2y)=\log x+C---(2)`

If x = 0, y = 0

`\implies C=0`

From equation (2),

`(1)/(2) \log(1+2y)=log x`

`\log (1+2y) = 2\log x`

`\log (1+2y) = \log x^2`

`\implies 1 + 2y = x^2`