Find the general solution

**matty888** · May 15th 2009, 02:00 AM

Find the general solution to the first order linear diff. equation

dy/dx + 3xy = -2x

**Soroban** · May 15th 2009, 02:26 AM

Hello, matty888!

Find the general solution of: . $<br /> \frac{dy}{dx} + 3xy \:=\: -2x$

Integrating factor: . $I \;=\;e^{\int3x\,dx} \;=\;e^{\frac{3}{2}x^2}$

Multiply: . $e^{\frac{3}{2}x^2}\frac{dy}{dx} + 3xye^{\frac{3}{2}x^2} \;=\;-2xe^{\frac{3}{2}x^2}$

We have: . $\frac{d}{dx}\left(e^{\frac{3}{2}x^2}y\right) \;=\;-2xe^{\frac{3}{2}x^2}$

Integrate: . $e^{\frac{3}{2}x^2}y \;=\;-2\int e^{\frac{3}{2}x^2}(x\,dx)$

On the right side:

. . Let: $u \:=\:\tfrac{3}{2}x^2 \quad\Rightarrow\quad du \:=\:3x\,dx \quad\Rightarrow\quad x\,dx \:=\:\tfrac{1}{3}du$

. . Substitute: . $-2\int e^u\left(\tfrac{1}{3}du\right) \;=\;-\tfrac{2}{3}\int e^u\,du \;=\;-\tfrac{2}{3}e^u + C$

. . Back-substitute: . $-\tfrac{2}{3}e^{\frac{3}{2}x^2} + C$

The equation becomes: . $e^{\frac{3}{2}x^2}y \;=\;-\tfrac{2}{3}e^{\frac{3}{2}x^2} + C$

Therefore: . $\boxed{y \;=\;-\frac{2}{3} + Ce^{-\frac{3}{2}x^2}}$

**matty888** · May 15th 2009, 02:50 AM

Hi Soroban,
Great answer,very clear & helpful,many thanks!!
matty888

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Find the general solution

Thanks Soroban

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