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Solve the following homogeneous differential equation : \[ x \frac{d y}{d x}=x+y \] |
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Answer» x \(\frac{dy}{dx}\) = x + y ⇒ \(\frac{dy}{dx}\) = \(\frac{x+y}x\) = 1 + y/x. Let y/x = v ⇒ y = v x ⇒ \(\frac{dy}{dx}\) = v + x \(\frac{dv}{dx}\) Then, v + x \(\frac{dv}{dx}\) = 1 + v ⇒ d v = dx/x ⇒ v = ln x + ln c, where ln c is integral constant. ⇒ y/x = ln cx (∵ ln A + ln B = ln AB) ⇒ y = x ln cx which is a solution of given differential equation. |
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