The function is the solution of the differential equation in satisfying . Then is

Question

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

Smart Achievers · Accepted Answer

(2)

$\frac{d y}{d x} + \frac{x}{x^{2} - 1} y = \frac{x^{4} + 2 x}{\sqrt{1 - x^{2}}}$

$The I.F. of this differential equation is$ $e^{\int \frac{x}{x^{2} - 1} d x} = e^{- \int \frac{x}{1 - x^{2}} d x} = e^{\frac{1}{2} \log (1 - x^{2})} = \sqrt{1 - x^{2}}$

$The solution is given by$ $y \sqrt{1 - x^{2}} = \int \frac{x^{3} + 2}{\sqrt{1 - x^{2}}} \sqrt{1 - x^{2}} d x + λ$

$= \int (x^{4} + 2 x) d x + λ$ $= \frac{x^{5}}{5} + x^{2} + λ$

$At y (0) = 0 \Rightarrow λ = 0$ $\Rightarrow y \sqrt{1 - x^{2}} = \frac{x^{5}}{5} + x^{2}$

$\int_{- \sqrt{3} / 2}^{\sqrt{3} / 2} \frac{\frac{x^{5}}{5} + x^{2}}{\sqrt{1 - x^{2}}} d x = \int_{- \sqrt{3} / 2}^{\sqrt{3} / 2} \frac{x^{2}}{\sqrt{1 - x^{2}}} d x$

$(The other part is odd)$ $= 2 \int_{0}^{\sqrt{3} / 2} \frac{x^{2}}{\sqrt{1 - x^{2}}} d x$

$Let x = \sin θ, we get$

$I = 2 \int_{0}^{π / 3} \frac{\sin^{2} θ}{\cos θ} \cos θ d θ$

$= 2 \int_{0}^{π / 3} \sin^{2} θ d θ$

$= \int_{0}^{π / 3} (1 - \cos 2 θ) d θ$

$= {[θ - \frac{\sin 2 θ}{2}]}_{0}^{π / 3}$

$= \frac{π}{3} - \frac{\sqrt{3}}{4}$

$∴$ $2 I = \frac{2 π}{3} - \frac{\sqrt{3}}{2}$

Solution

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

1 $\frac{2 π}{3} + \frac{\sqrt{3}}{2}$

2 $\frac{2 π}{3} - \frac{\sqrt{3}}{2}$

3 $\frac{π}{6} - \frac{\sqrt{3}}{4}$

4 $\frac{π}{6} - \frac{\sqrt{3}}{2}$

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Solution

Q. The function y=f(x) is the solution of the differential equation dydx+xyx2-1=x4+21-x2 in (-1,1) satisfying f(0)=0. Then 2∫-3/23/2f(x) dx is

1 2π3+32

2 2π3-32

3 π6-34