First slide

Methods of solving first order first degree differential equations

Question

The general solution of $y' + x y = 4 x$ is given by

Moderate

A

$y = e^{(- 1 / 2) x^{2}} + C x$
B

$y = e^{(1 / 2) x^{2}} + C x$
C

$y = C e^{(1 / 2) x^{2}} - 4$
D

$y = e^{(1 / 2) x^{2}} + C x + 4$

Solution

The given equation is a linear equation with $I . F . = e^{- \int xdx} = e^{- \frac{1}{2} x^{2}}$ .
so $\begin{array}{l} \frac{d}{d x} (y e^{- \frac{1}{2} x^{2}}) = 4 x e^{- \frac{1}{2} x^{2}} \\ \Rightarrow y e^{- \frac{1}{2} x^{2}} = 4 \int x e^{- \frac{1}{2} x^{2}} d x + C = - 4 e^{- \frac{1}{2} x^{2}} + C \end{array}$
so $y = - 4 + C e^{\frac{1}{2} x^{2}}$

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