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Misha Larkins [42]
3 years ago
12

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

Mathematics
1 answer:
JulijaS [17]3 years ago
8 0

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

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