Question

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

Answer 1

Answer:

$1 + 2y = x^2$

Step-by-step explanation:

Given differential equation,

$\frac{dy}{dx}-2xy=x$

$\frac{dy}{dx}=x+2xy$

$\frac{dy}{dx}=x(1+2y)$

$\frac{dy}{1+2y}=xdx$

Integrating both sides,

$\int \frac{dy}{1+2y}=\int xdx----(1)$

Put 1 + 2y = u

Differentiating both sides,

2dy = du

$\implies dy=\frac{du}{2}$

From equation (1),

$\frac{1}{2} \int \frac{du}{u}=\int xdx$

$\frac{1}{2}\log u = \log x + C$

$\frac{1}{2} \log (1+2y)=\log x+C---(2)$

If x = 0, y = 0

$\implies C=0$

From equation (2),

$\frac{1}{2} \log(1+2y)=log x$

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

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

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