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Yukti Vault

Calculus

Differential Equation: JEE Main & Advanced PYQs

313 previous year questions on Differential Equation from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 51-100 of 313

  1. Let $y=y(x)$ be a solution curve of the differential equation $(y+1)\tan^{2}x\,dx+\tan x\,dy+y\,dx=0$, $x\in\left(0,\frac{\pi}{2}\right)$. If $\lim_{x\to0+}xy(x)=1$ …
  2. If a curve $y=y(x)$ passes through the point $\left(1,\frac{\pi}{2}\right)$ and satisfies the differential equation …
  3. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2\sin x-5)x-2\cos x$ such that $y(0)=7$. Then $y(\pi)$ is equal to :
  4. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=1+xe^{y-x}$, $-\sqrt{2}<x<\sqrt{2}$, $y(0)=0$ then, the minimum value of $y(x)$ …
  5. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x\tan x)}$, $x\in\left(0,\frac{\pi}{2}\right)$ satisfying …
  6. If $\frac{dy}{dx}=\frac{2^{x+y}-2^{x}}{2^{y}}$, $y(0)=1$, then $y(1)$ is equal to :
  7. Let $y=y(x)$ be the solution of the differential equation $(2x\log_ex)\frac{dy}{dx}+2y=\frac{3}{x}\log_ex$, $x>0$ and $y\left(e^{-1}\right)=0$. Then $y(e)$ is equal to
  8. Let $g$ be a differentiable function such that $\int_{0}^{x}g(t)\,dt=x-\int_{0}^{x}tg(t)\,dt$, $x\ge0$ and let $y=y(x)$ satisfy the differential equation …
  9. Let $y=y(x)$ be the solution of the differential equation $\left(1+y^{2}\right)e^{\tan x}dx+\cos^{2}x\left(1+e^{2\tan x}\right)dy=0$, $y(0)=1$. Then …
  10. Let $f:[0,\infty)\to\mathbb{R}$ be a differentiable function such that $f(x)=1-2x+\int_{0}^{x}e^{x-t}f(t)\,dt$ for all $x\in[0,\infty)$. Then the area of the region …
  11. Suppose the solution of the differential equation $\frac{dy}{dx}=\frac{(2+\alpha)x-\beta y+2}{\beta x-2\alpha y-(\beta\gamma-4\alpha)}$ represents a circle passing …
  12. Let $f(x)=x-1$ and $g(x)=e^{x}$ for $x\in\mathbb{R}$. If $\frac{dy}{dx}=\left(e^{-2\sqrt{x}}g(f(f(x)))-\frac{y}{\sqrt{x}}\right)$, $y(0)=0$, then $y(1)$ is
  13. If $y\frac{dy}{dx}=x\left[\frac{y^{2}}{x^{2}}+\frac{\phi\left(\frac{y^{2}}{x^{2}}\right)}{\phi'\left(\frac{y^{2}}{x^{2}}\right)}\right]$, $x>0$, $\phi>0$, and $y(1)=-1$ …
  14. Let $\alpha$ be a non-zero real number. Suppose $f:R\to R$ is a differentiable function such that $f(0)=1$ and $\lim_{x\to-\infty}f(x)=1$. If $f'(x)=\alpha f(x)+3$, for …
  15. A differential equation representing the family of parabolas with axis parallel to $y$-axis and whose length of latus rectum is the distance of the point $(2,-3)$ from …
  16. A function $y=f(x)$ satisfies $f(x)\sin2x+\sin x-\left(1+\cos^{2}x\right)f'(x)=0$ with condition $f(0)=0$. Then $f\left(\frac{\pi}{2}\right)$ is equal to
  17. If the solution of the differential equation $\frac{dy}{dx}+e^{x}\left(x^{2}-2\right)y=\left(x^{2}-2x\right)\left(x^{2}-2\right)e^{2x}$ satisfies $y(0)=0$, then the …
  18. If $x=x(y)$ is the solution of the differential equation $y\frac{dx}{dy}=2x+y^{3}(y+1)e^{y}$, $x(1)=0$; then $x(e)$ is equal to :
  19. Let the solution curve $y=f(x)$ of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$, $x\in(-1,1)$ pass through the origin. Then …
  20. Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+\frac{1}{x^2-1}y=\left(\frac{x-1}{x+1}\right)^{1/2}$, $x>1$ passing through the point …
  21. Let the solution curve $y=y(x)$ of the differential equation $(1+e^{2x})\left(\frac{dy}{dx}+y\right)=1$ pass through the point $\left(0,\frac\pi2\right)$. Then …
  22. The differential equation of the family of curves, $x^2=4b(y+b)$, $b\in\mathbf{R}$, is :
  23. If the solution curve of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passes through the points $(2,1)$ and $(k+1,2)$, $k>0$, then
  24. If $x^3dy+xy\,dx=x^2dy+2y\,dx$; $y(2)=e$ and $x>1$, then $y(4)$ is equal to :
  25. If a curve $y=f(x)$, passing through the point $(1,2)$, is the solution of the differential equation, $2x^2dy=(2xy+y^2)dx$, then $f\left(\frac12\right)$ is equal to :
  26. Let $y=y(x)$ be the solution of the differential equation, $\frac{2+\sin x}{y+1}\cdot\frac{dy}{dx}=-\cos x$, $y>0$, $y(0)=1$. If $y(\pi)=a$ and $\frac{dy}{dx}$ at …
  27. Let the solution curve $y=y(x)$ of the differential equation …
  28. The differential equation of the family of circles passing through the points $(0,2)$ and $(0,-2)$ is :
  29. If $y=y(x)$ is the solution of the differential equation $x\frac{dy}{dx}+2y=xe^{x}$, $y(1)=0$, then the local maximum value of the function $z(x)=x^{2}y(x)-e^{x}$ …
  30. Let $x=x(y)$ be the solution of the differential equation $2ye^{x/y^{2}}dx+\left(y^{2}-4xe^{x/y^{2}}\right)dy=0$ such that $x(1)=0$. Then, $x(e)$ is equal to :
  31. Let $y=y(x)$ be the solution of the differential equation $(\tan x)^{1/2}dy=\left(\sec^{3}x-(\tan x)^{3/2}y\right)dx$, $0<x<\frac{\pi}{2}$ …
  32. If the solution curve of the differential equation $\left((\tan^{-1}y)-x\right)dy=\left(1+y^{2}\right)dx$ passes through the point $(1,0)$, then the abscissa of the …
  33. If $y=y(x)$ is the solution of the differential equation $\frac{5+e^x}{2+y}\cdot\frac{dy}{dx}+e^x=0$ satisfying $y(0)=1$, then a value of $y(\log_e13)$ is :
  34. If $y=y(x)$ satisfies the differential equation $x\,dy+\left(x\tan\left(\frac yx\right)-y\right)dx=0$, $x>0$, and $y(1)=\frac\pi2$, then a value of $y(\sqrt2)$ is
  35. If $\frac{dy}{dx}+\frac{2^{x-y}\left(2^{y}-1\right)}{2^{x}-1}=0$, $x,y>0$, $y(1)=1$, then $y(2)$ is equal to :
  36. If $y=y(x)$ is the solution of the differential equation $2x^{2}\frac{dy}{dx}-2xy+3y^{2}=0$ such that $y(e)=\frac{e}{3}$, then $y(1)$ is equal to
  37. Consider a curve $y=y(x)$ in the first quadrant as shown in the figure [figure not transcribed: the curve passes through the point $(4,2)$ on the $x$-axis side marked …
  38. If $\frac{dy}{dx}+2y\tan x=\sin x$, $0<x<\frac\pi2$ and $y\left(\frac\pi3\right)=0$, then the maximum value of $y(x)$ is :
  39. Let the slope of the tangent to a curve $y=f(x)$ at $(x,y)$ be given by $2\tan x(\cos x-y)$. If the curve passes through the point $\left(\frac{\pi}{4},0\right)$, then …
  40. Let $y=y(x)$ be the solution of the differential equation $x\left(1-x^{2}\right)\frac{dy}{dx}+\left(3x^{2}y-y-4x^{3}\right)=0$, $x>1$, with $y(2)=-2$. Then $y(3)$ is …
  41. Let $g:(0,\infty)\to\mathbf{R}$ be a differentiable function such that …
  42. The slope of the tangent to a curve $C:y=y(x)$ at any point $(x,y)$ on it is $\frac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}$. If $C$ passes through the points …
  43. If $y=y(x)$ is the solution of the differential equation $\left(1+e^{2x}\right)\frac{dy}{dx}+2\left(1+y^{2}\right)e^{x}=0$ and $y(0)=0$, then …
  44. If $y=y(x)$, $x\in\left(0,\frac\pi2\right)$ be the solution curve of the differential equation $(\sin^22x)\frac{dy}{dx}+(8\sin^22x+2\sin4x)y=2e^{-4x}(2\sin2x+\cos2x)$ …
  45. The general solution of the differential equation $(x-y^2)dx+y(5x+y^2)dy=0$ is :
  46. The solution curve of the differential equation, $(1+e^{-x})(1+y^2)\frac{dy}{dx}=y^2$, which passes through the point $(0,1)$, is :
  47. If $y=\left(\frac2\pi x-1\right)\mathrm{cosec}\,x$ is the solution of the differential equation, $\frac{dy}{dx}+p(x)y=\frac2\pi\mathrm{cosec}\,x$, $0<x<\frac\pi2$, then …
  48. Let $\frac{dy}{dx}=\frac{ax-by+a}{bx+cy+a}$, $a,b,c\in\mathbf{R}$, represents a circle with center $(\alpha,\beta)$. Then $\alpha+2\beta$ is equal to :
  49. If $\frac{dy}{dx}=\frac{xy}{x^2+y^2}$; $y(1)=1$; then a value of $x$ satisfying $y(x)=e$ is :
  50. Let $y=y(x)$ be the solution of the differential equation, $xy'-y=x^2(x\cos x+\sin x)$, $x>0$. If $y(\pi)=\pi$, then $y''\left(\frac\pi2\right)+y\left(\frac\pi2\right)$ …

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