First slide

Types of solutions of differential equations

Question

$The solution of the differential equation x^{3} \frac{dy}{dx} + 4 x^{2} tany = e^{x} secy satisfying y (1) =0 is$

Moderate

A

${tany=(x-2)e}^{x} logx$
B

${siny=e}^{x} {(x-1)x}^{-4}$
C

${tany=(x-1)e}^{x} x^{-3}$
D

${siny=e}^{x} {(x-1)x}^{-3}$

Solution

$\begin{array}{l} x^{4} cosydy + 4 x^{3} \sin y = x e^{x} \\ \to \int d (x^{4} siny) = \int x e^{x} d x \\ \Rightarrow x^{4} \sin y = e^{x} (x - 1) + c - \dots - - - 1 \\ Pass (1, 0) \Rightarrow c = 0 \\ \to x^{4} \sin y = e^{x} (x - 1) \\ \to \sin y = e^{x} (x - 1) x^{4} \end{array}$

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