Question

if dx
dy
y e cos x sin x find ax 3 2 

Answer 1

Answer on Question #62565 – Math – Differential Equations

Question

If $y = e^{ax} \cdot \cos^3 x \cdot \sin^2 x$, find $\frac{dy}{dx}$

Solution

$$\begin{aligned} \frac{dy}{dx} = \frac{d}{dx} \left( e^{ax} \cdot \cos^3 x \cdot \sin^2 x \right) &= \frac{de^{ax}}{dx} \cos^3 x \sin^2 x + e^{ax} \frac{d \cos^3 x}{dx} \sin^2 x + e^{ax} \cos^3 x \frac{d \sin^2 x}{dx} \\ &= e^{ax} (ax)' \cos^3 x \sin^2 x + e^{ax} \left( 3 \cos^2 x (\cos x)' \sin^2 x + e^{ax} \cos^3 x \right) 2 \sin x (\sin x)' \\ &= a e^{ax} \cos^3 x \sin^2 x - 3 e^{ax} \cos^2 x \sin^3 x + 2 e^{ax} \cos^4 x \sin x \\ &= e^{ax} \cos^2 x \sin x \left( a \cos x \sin x - 3 \sin^2 x + 2 \cos^2 x \right) \\ &= \frac{1}{2} e^{ax} \cos x \sin 2x \left( \frac{a}{2} \sin 2x - \frac{3}{2} (1 - \cos 2x) + (1 + \cos 2x) \right) = \\ &= \frac{1}{4} e^{ax} \cos x \sin 2x \left( a \sin 2x + 5 \cos 2x - 1 \right) \end{aligned}$$

Answer: $\frac{dy}{dx} = \frac{1}{4} e^{ax} \cos x \sin 2x$ ($a \sin 2x + 5 \cos 2x - 1$).

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