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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 101-150 of 313

  1. Let $y=y(x)$ be a solution of the differential equation, $\sqrt{1-x^2}\frac{dy}{dx}+\sqrt{1-y^2}=0$, $|x|<1$. If $y\left(\frac12\right)=\frac{\sqrt3}{2}$, then …
  2. Let $f$ be a twice differentiable function such that $f(x)=\int_{0}^{x}\tan(t-x)\,dt-\int_{0}^{x}f(t)\tan t\,dt$, $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Then …
  3. Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+\left(\frac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)y=\frac{x+3}{x+1}$, $x>-1$, which passes …
  4. Let $\frac{dy}{dx}=\frac{ax-by+a}{bx+cy+a}$, where $a,b,c$ are constants, represent a circle passing through the point $(2,5)$. Then the shortest distance of the point …
  5. The slope of normal at any point $(x,y)$, $x>0$, $y>0$ on the curve $y=y(x)$ is given by $\frac{x^{2}}{xy-x^{2}y^{2}-1}$. If the curve passes through the point $(1,1)$ …
  6. Let $y=y(x)$ be solution of the differential equation $\log_e\left(\frac{dy}{dx}\right)=3x+4y$, with $y(0)=0$. If $y\left(-\frac23\log_e2\right)=\alpha\log_e2$, then the …
  7. Let $y=y_1(x)$ and $y=y_2(x)$ be two distinct solutions of the differential equation $\frac{dy}{dx}=x+y$, with $y_1(0)=0$ and $y_2(0)=1$ respectively. Then, the number …
  8. If the solution curve $y=y(x)$ of the differential equation $y^{2}dx+\left(x^{2}-xy+y^{2}\right)dy=0$, which passes through the point $(1,1)$ and intersects the line …
  9. Let the solution curve of the differential equation $x\frac{dy}{dx}-y=\sqrt{y^{2}+16x^{2}}$, $y(1)=3$ be $y=y(x)$. Then $y(2)$ is equal to :
  10. If the curve, $y=y(x)$ represented by the solution of the differential equation $\left(2xy^{2}-y\right)dx+x\,dy=0$, passes through the intersection of the lines …
  11. The solution of the differential equation $\frac{dy}{dx}-\frac{y+3x}{\log_e(y+3x)}+3=0$ is : (where $C$ is a constant of integration.)
  12. Let $y=y(x)$ be the solution of the differential equation $(x-x^3)dy=(y+yx^2-3x^4)dx$, $x>2$. If $y(3)=3$, then $y(4)$ is equal to :
  13. Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x,y)$ on it is directly proportional to $\left(\frac{-y}{x}\right)$. If the curve passes …
  14. Let the solution curve of the differential equation $x\,dy=\left(\sqrt{x^2+y^2}+y\right)dx$, $x>0$, intersect the line $x=1$ at $y=0$ and the line $x=2$ at $y=\alpha$. …
  15. Let $y=y(x)$ be the solution of the differential equation $(x+1)y'-y=e^{3x}(x+1)^{2}$, with $y(0)=\frac{1}{3}$. Then, the point $x=-\frac{4}{3}$ for the curve $y=y(x)$ …
  16. If $x\phi(x)=\int_{5}^{x}\left(3t^{2}-2\phi'(t)\right)dt$, $x>-2$, and $\phi(0)=4$, then $\phi(2)$ is $\_\_\_\_\_\_$ .
  17. Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the $x$-axis and between the abscissae $3$ and $x(>3)$ be …
  18. Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y-y=Y'(x)(X-x)$ and the co-ordinate axes, where $(x,y)$ is any point on the …
  19. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+2y\sec^{2}x=2\sec^{2}x+3\tan x\cdot\sec^{2}x$ such that $y(0)=\frac{5}{4}$. Then …
  20. For a differentiable function $f:\mathbb{R}\to\mathbb{R}$, suppose $f'(x)=3f(x)+\alpha$, where $\alpha\in\mathbb{R}$, $f(0)=1$ and $\lim_{x\to-\infty}f(x)=7$. Then …
  21. If for $x\ge0$, $y=y(x)$ is the solution of the differential equation, $(x+1)dy=((x+1)^2+y-3)dx$, $y(2)=0$, then $y(3)$ is equal to ________.
  22. Let $y=y(x)$ be the solution curve of the differential equation $\sin(2x^2)\log_e(\tan x^2)\,dy+\left(4xy-4\sqrt2x\sin\left(x^2-\frac\pi4\right)\right)dx=0$ …
  23. Let $\alpha|x|=|y|e^{xy-\beta}$, $\alpha,\beta\in\mathbb{N}$ be the solution of the differential equation $x\,dy-y\,dx+xy(x\,dy+y\,dx)=0$, $y(1)=2$. Then $\alpha+\beta$ …
  24. Let $y=y(x)$ be the solution of the differential equation $\left(1-x^{2}\right)dy=\left(xy+\left(x^{3}+2\right)\sqrt{1-x^{2}}\right)dx$, $-1<x<1$, and $y(0)=0$. If …
  25. Let a curve $y=f(x)$ pass through the point $\left(2,(\log_e2)^{2}\right)$ and have slope $\frac{2y}{x\log_ex}$ for all positive real value of $x$. Then the value of …
  26. Let $y=y(x)$ be the solution of the differential equation $\sec^{2}x\,dx+\left(e^{2y}\tan^{2}x+\tan x\right)dy=0$, $0<x<\frac{\pi}{2}$, $y\left(\frac{\pi}{4}\right)=0$. …
  27. The difference between degree and order of a differential equation that represents the family of curves given by $y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right)$, $a>0$ is …
  28. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{4y^3+2yx^2}{3xy^2+x^3}$, $y(1)=1$. If for some $n\in\mathbb{N}$, $y(2)\in[n-1,n)$, then …
  29. Let $y=y(x)$ be the solution of the differential equation $\left(1-x^{2}\right)dy=\left[xy+\left(x^{3}+2\right)\sqrt{3\left(1-x^{2}\right)}\right]dx$, $-1<x<1$ …
  30. If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{4x}{\left(x^{2}-1\right)}y=\frac{x+2}{\left(x^{2}-1\right)^{5/2}}$, $x>1$ such that …
  31. If the solution curve $y=y(x)$ of the differential equation $\left(1+y^{2}\right)(1+\log_ex)dx+x\,dy=0$, $x>0$ passes through the point $(1,1)$ and …
  32. Suppose $y=y(x)$ be the solution curve to the differential equation $\frac{dy}{dx}-y=2-e^{-x}$ such that $\lim_{x\to\infty}y(x)$ is finite. If $a$ and $b$ are …
  33. Let $S=(0,2\pi)-\left\{\frac{\pi}{2},\frac{3\pi}{4},\frac{3\pi}{2},\frac{7\pi}{4}\right\}$. Let $y=y(x)$, $x\in S$, be the solution curve of the differential equation …
  34. Let the solution curve $y=y(x)$ of the differential equation $\left(4+x^{2}\right)dy-2x\left(x^{2}+3y+4\right)dx=0$ pass through the origin. Then $y(2)$ is equal to …
  35. If the solution $y=y(x)$ of the differential equation $\left(y+3x^{4}\right)\frac{dx}{dy}=x$, $x>0$, satisfies $y(1)=-1$, then the value of …
  36. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{\sqrt{2}y}{2\cos^{4}x-\cos2x}=xe^{\tan^{-1}(\sqrt{2}\cot2x)}$, $0<x<\frac{\pi}{2}$ with …
  37. Let a curve $y=f(x)$, $x\in(0,\infty)$ pass through the points $P\left(1,\frac{3}{2}\right)$ and $Q\left(a,\frac{1}{2}\right)$. If the tangent at any point $R(b,f(b))$ …
  38. If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0$, $y(0)=3$, is $\alpha x+\beta y+3\log_e|2x+3y-\gamma|=6$, then $\alpha+2\beta+3\gamma$ is equal …
  39. Let $y=y(x)$ be a solution of the differential equation $(x\cos x)dy+(xy\sin x+y\cos x-1)dx=0$, $0<x<\frac{\pi}{2}$. If …
  40. Let $y=y(x)$ be the solution of the differential equation $x\,dy-y\,dx=\sqrt{\left(x^{2}-y^{2}\right)}\,dx$, $x\ge1$, with $y(1)=0$. If the area bounded by the line …
  41. If $y^{1/4}+y^{-1/4}=2x$, and $\left(x^{2}-1\right)\frac{d^{2}y}{dx^{2}}+\alpha x\frac{dy}{dx}+\beta y=0$, then $|\alpha-\beta|$ is equal to $\_\_\_\_\_\_$ .
  42. If $y=y(x)$, $y\in\left[0,\frac\pi2\right)$ is the solution of the differential equation $\sec y\frac{dy}{dx}-\sin(x+y)-\sin(x-y)=0$, with $y(0)=0$, then …
  43. A hostel has $100$ students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus …
  44. If $x=x(t)$ is the solution of the differential equation $(t+1)dx=\left(2x+(t+1)^{4}\right)dt$, $x(0)=2$, then, $x(1)$ equals $\_\_\_\_\_\_$ .
  45. If the solution curve of the differential equation $\left(y-2\log_ex\right)dx+\left(x\log_ex^{2}\right)dy=0$, $x>1$ passes through the points …
  46. If the solution $y(x)$ of the given differential equation $\left(e^{y}+1\right)\cos x\,dx+e^{y}\sin x\,dy=0$ passes through the point $\left(\frac{\pi}{2},0\right)$ …
  47. Let $f:\mathbf{R}\to\mathbf{R}$ be a thrice differentiable odd function satisfying $f'(x)\ge0$, $f''(x)=f(x)$, $f(0)=0$, $f'(0)=3$. Then $9f(\log_e3)$ is equal to …
  48. If $y=y(x)$ is the solution of the equation $e^{\sin y}\cos y\frac{dy}{dx}+e^{\sin y}\cos x=\cos x$, $y(0)=0$; then …
  49. Let $y=y(x)$, $x>1$, be the solution of the differential equation $(x-1)\frac{dy}{dx}+2xy=\frac{1}{x-1}$, with $y(2)=\frac{1+e^{4}}{2e^{4}}$. If …
  50. Let the solution curve $x=x(y)$, $0<y<\frac{\pi}{2}$, of the differential equation …

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