First slide

Methods of solving first order first degree differential equations

Question

$The function y = f (x) is the solution of the differential equation$ $\frac{dy}{dx} + \frac{xy}{x^{2} - 1} = \frac{x^{4} + 2 x}{\sqrt{1 - x^{2}}} in x \in (- 1, 1) satisfying f (0) = 0 the \int_{- \frac{\sqrt{3}}{2}}^{\sqrt{3} / 2} f (x) d x is$

Moderate

A

$\frac{π}{3} - \frac{\sqrt{3}}{2}$
B

$\frac{π}{3} - \frac{\sqrt{3}}{4}$
C

$\frac{π}{6} - \frac{\sqrt{3}}{4}$
D

$\frac{π}{6} - \frac{\sqrt{3}}{2}$

Solution

$I.F.= e^{\int \frac{x}{x^{2} - 1} dx} = e^{\frac{1}{2} \log (x^{2} - 1) ∣} = \sqrt{1 - x^{2}}$
$Solution of D.E is y \sqrt{1 - x^{2}} = \int \frac{x^{4} + 2 x}{\sqrt{1 - x^{2}}} \cdot \sqrt{1 - x^{2}} d y = \frac{x^{5}}{5} + x^{2} + c$
$\begin{array}{l} f (0) = 0 \Rightarrow c = 0 \\ y \sqrt{1 - x^{2}} = \frac{x^{5}}{5} + x^{2} \\ \int_{- \frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} y dx = \int_{- \frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} \frac{x^{2}}{\sqrt{1 - x^{2}}} dx \\ I f f (- x) = f (x) t h e n \int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x \\ = 2 \int_{0}^{\frac{\sqrt{3}}{2}} \frac{x^{2}}{\sqrt{1 - x^{2}}} dx, put x = \sin θ \Rightarrow d x = \cos θ d θ \\ = 2 \int_{0}^{π / 3} \sin^{2} θ d θ \\ = 2 \int \frac{1 - \cos 2 θ}{2} d θ \\ = {(θ - \frac{\sin 2 θ}{2})}_{0}^{\frac{π}{3}} \\ = \frac{π}{3} - \frac{\sqrt{3}}{4} \end{array}$

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