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JEE MATHEMATICS MCQ: DIFFERENTIAL EQUATIONS

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The solution of the differential equation xdx+ydyxdx−ydy=y3x3\frac{\sqrt{x} d x+\sqrt{y} d y}{\sqrt{x} d x-\sqrt{y} d y}=\sqrt{\frac{y^3}{x^3}} is given by,
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Correct option: D
12log⁡(x3+y3)+tan⁡−1(yx)3/2=c\frac{1}{2} \log \left(x^3+y^3\right)+\tan ^{-1}\left(\frac{y}{x}\right)^{3 / 2}=c
Option D (12log⁡(x3+y3)+tan⁡−1(yx)3/2=c\frac{{1}}{{2}} \log \left(x^3+y^3\right)+\tan ^{{-1}}\left(\frac{{y}}{{x}}\right)^{{3 / 2}}=c) is correct because it represents the solution to the given differential equation, obtained through homogeneous substitution and integration.

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Let y=y(x)y=y(x) be the solution of the differential equation (3y2−5x2)ydx+2x(x2−y2)dy=0\left(3 y^{2}-5 x^{2}\right) y d x+2 x\left(x^{2}-y^{2}\right) d y=0 such that y(1)=1y(1)=1. Then ∣(y(2))3−12y(2)∣\left|(y(2))^{3}-12 y(2)\right| is equal to : The degree and order of the differential equation whose solution is a parabola whose axis is xx-axis, are If x2+y2=1{x^2} + {y^2} = 1, then(y′=dydx,y′′=d2ydx2)\left( {y' = \frac{{dy}}{{dx}},y'' = \frac{{{d^2}y}}{{d{x^2}}}} \right) If x=Acos⁡4t+Bsin⁡4t,x = A\cos 4t + B\sin 4t, then d2xdt2\frac{{{d^2}x}}{{d{t^2}}} is equal to Let y=y1(x)y=y_{1}(x) and y=y2(x)y=y_{2}(x) be the solution curves of the differential equation dydx=y+7\frac{d y}{d x}=y+7 with initial conditions y1(0)=0y_{1}(0)=0 and y2(0)=1y_{2}(0)=1 respectively. Then the curves y=y1(x)y=y_{1}(x) and y=y2(x)y=y_{2}(x) intersect at Let y=y(t)y=y(t) be a solution of the differential equation dydt+αy=γe−βt\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}
where, α>0,β>0\alpha>0, \beta>0 and γ>0\gamma>0. Then lim⁡x→∞y(t)\lim \limits_{{x \rightarrow \infty}} y(t)
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