First slide

Methods of solving first order first degree differential equations

Question

$Let y = y (x) be the solution of the D.E \sin x \frac{d y}{d x} + y \cos x = 4 x, x \in (0, π) . If$ $\Rightarrow y (π / 2) = 0 then y (π / 6) is$

Moderate

A

$- \frac{4}{9} π^{2}$
B

$\frac{4}{9 \sqrt{3}} π^{2}$
C

$- \frac{8}{9 \sqrt{3}} π^{2}$
D

$- \frac{8}{9} π^{2}$

Solution

$\begin{array}{l} \int d (ysinx) = \int 4 x \\ \Rightarrow ysinx = 2 x^{2} + c passes through (\frac{π}{2}, 0) \\ \Rightarrow c = - \frac{π^{2}}{2} \\ ysinx = 2 x^{2} - \frac{π^{2}}{2} \Rightarrow put x = \frac{π}{6} \\ \Rightarrow \frac{y}{2} = 2 \cdot \frac{π^{2}}{36} - \frac{π^{2}}{2} \Rightarrow y = - \frac{8 π^{2}}{9} \end{array}$

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