Question

solve the differential equation
dy/dx + 2y/x =x2

Answer 1

Do you know what's the integrating factor here?

Answer 2

This is a first-order linear ODE

Answer 3

So,
\[\frac{dy}{dx}+P(x)y=Q(x)\]
Integrating factor
\[\huge \mu(x)=e^{\int\limits P(x)dx}\]

Answer 4

you doing better

Answer 5

So what's P(x)?

Answer 6

2/x

Answer 7

Yes so,
\[\Large \mu(x)=e^{\int\limits \frac{2}{x}dx}=e^{2lnx}=x^2\]
Now multiply \(\mu(x)\) across
\[\frac{dy}{dx} + \frac{2y}{x} =x^2 \\ \\ x^2\frac{dy}{dx} + 2xy=x^4\]
Then use reverse product rule
\[\frac{d}{dx}(x^2y)=x^4\]
Integrate both sides and you're done

Answer 8

please show me last resultant

Answer 9

Just integrate
\[\int\limits \frac{d}{dx}(x^2y)dx=\int\limits x^4 dx \\ \\ x^2y=\frac{x^5}{5}+c_1\]
\[y=\frac{x^3}{5}+\frac{c_1}{x^2}\]

Answer 10

thank you very much

Answer 11

yw