First slide

Methods of solving first order first degree differential equations

Question

$If y + x \frac{dy}{dx} = x \frac{φ (xy)}{φ^{1} (xy)} then φ (xy) is$

Moderate

A

${Ke}^{x^{2} /2}$
B

${Ke}^{y^{2} /2}$
C

${Ke}^{xy/2}$
D

${Ke}^{xy}$

Solution

$\begin{array}{l} Put xy = v \Rightarrow y + x \frac{dy}{dx} = \frac{dv}{dx} \\ \frac{dv}{dx} = x \frac{φ (v)}{φ^{1} (v)} \\ \int \frac{φ^{1} (v)}{φ (v)} dv = \int xdx \\ \ln | φ (v) | = \frac{x^{2}}{2} + \ln c \\ φ (v) = {Ke}^{\frac{x^{2}}{2}} \\ φ (xy) = {Ke}^{\frac{x^{2}}{2}} \end{array}$

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