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Question: The general solution of y. \(\frac{d^{2}y}{dx^{2}}\) = \(\left( \frac{dy}{dx} \right)^{2}\) is –...

The general solution of y. d2ydx2\frac{d^{2}y}{dx^{2}} = (dydx)2\left( \frac{dy}{dx} \right)^{2} is –

A

y = C1x + C2

B

y = C1eC2xe^{C_{2}x}

C

y = C1x + C2ex

D

y = eC1xe^{C_{1}x}+eC2xe^{C_{2}x}

Answer

y = C1eC2xe^{C_{2}x}

Explanation

Solution

∫d(dydx)dydx\int_{}^{}\frac{d\left( \frac{dy}{dx} \right)}{\frac{dy}{dx}} = ∫dyy\int_{}^{}\frac{dy}{y}

\ ln (dydx)\left( \frac{dy}{dx} \right) = lny + C

\ 1ydydx\frac{1}{y}\frac{dy}{dx} = C1 i.e. dyy\frac{dy}{y} = C1 dx

lny = C1x + C2

\ y = eC1x+C2e^{C_{1}x + C_{2}} = keC1xke^{C_{1}x}