The function $$y=f(x)$$ is the solution of the differential equation $$dydx+xyx2$$−$$1=x4+2x1$$−$$x2$$ in x∈$$($$−$$1$$,$$1)$$ satisfying $$f(0)=0$$ the ∫−$$323/2$$ $$f(x)dx$$ is

Question

The function $$y=f(x)$$ is the solution of the differential equation $$dydx+xyx2$$−$$1=x4+2x1$$−$$x2$$ in x∈$$($$−$$1$$,$$1)$$ satisfying $$f(0)=0$$ the ∫−$$323/2$$ $$f(x)dx$$ is

Infinity Learn · Accepted Answer

I.F.$$=$$ e∫$$xx2$$−$$1dx=e12log$$⁡$$x2$$−$$1$$∣$$=1$$−$$x2$$ Solution of D.E is $$y1$$−$$x2=$$∫$$x4+2x1$$−$$x2$$⋅$$1$$−$$x2dy=x55+x2+cf(0)=0$$⇒$$c=0y1$$−$$x2=x55+x2$$∫−$$3232$$ y $$dx=$$∫−$$3232$$ $$x21$$−$$x2dx$$     If $$f(-x)=f(x)$$ then ∫$$-aaf(x)$$ $$dx=2$$∫$$0af(x)$$ $$dx=2$$∫$$032$$ $$x21$$−$$x2dx$$, put $$x=$$$$sin$$⁡θ  ⇒$$dx=$$$$cos$$θ dθ $$=2$$∫$$0$$π$$/3$$ $$sin2$$⁡θ dθ$$=2$$∫$$1-cos2$$θ$$2d$$θ$$=$$θ$$-sin2$$θ$$20$$$$π$$$$3=$$π$$3-34$$

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

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The function y=f(x) is the solution of the differential equation dydx+xyx2−1=x4+2x1−x2 in x∈(−1,1) satisfying f(0)=0 the ∫−323/2 f(x)dx is

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