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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 1-50 of 313

  1. Let $f(x)=\int_{0}^{x}e^{t}f(t)\,dt+e^{x}$ be a differentiable function for all $x\in\mathbf{R}$. Then $f(x)$ equals :
  2. If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+(\tan x)y=\sin x$, $0\le x\le\frac{\pi}{3}$, with $y(0)=0$, then $y\left(\frac{\pi}{4}\right)$ …
  3. Let slope of the tangent line to a curve at any point $P(x,y)$ be given by $\frac{xy^{2}+y}{x}$. If the curve intersects the line $x+2y=4$ at $x=-2$, then the value of …
  4. If a curve passes through the origin and the slope of the tangent to it at any point $(x,y)$ is $\frac{x^{2}-4x+y+8}{x-2}$, then this curve also passes through the point …
  5. Let $y=y(x)$ be the solution of the differential equation $x\sqrt{1-x^{2}}\,dy+\left(y\sqrt{1-x^{2}}-x\cos^{-1}x\right)dx=0$, $x\in(0,1)$, $\lim_{x\to1^{-}}y(x)=1$. Then …
  6. Let $f:[1,\infty)\to\mathbf{R}$ be a differentiable function defined as $f(x)=\int_{1}^{x}f(t)\,dt+(1-x)(\log_e x-1)+e$. Then the value of $f(f(1))$ is :
  7. If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x\frac{dy}{dx}+y=bx^{4}$, then for what value of $b$, $\int_{1}^{2}f(x)dx=\frac{62}{5}$ ?
  8. If the curve $y=y(x)$ is the solution of the differential equation $2\left(x^{2}+x^{5/4}\right)dy-y\left(x+x^{1/4}\right)dx=2x^{9/4}dx$, $x>0$ which passes through the …
  9. Let $y=y(x)$ be the solution of the differential equation $\cos x(3\sin x+\cos x+3)dy=(1+y\sin x(3\sin x+\cos x+3))dx$, $0\le x\le\frac{\pi}{2}$, $y(0)=0$. Then …
  10. Let $C_1$ be the curve obtained by the solution of differential equation $2xy\frac{dy}{dx}=y^{2}-x^{2}$, $x>0$. Let the curve $C_2$ be the solution of …
  11. The population $P=P(t)$ at time '$t$' of a certain species follows the differential equation $\frac{dP}{dt}=0.5P-450$. If $P(0)=850$, then the time at which population …
  12. The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t=0$. The number of …
  13. Let $x=x(y)$ be the solution of the differential equation $2y^{2}\frac{dx}{dy}-2xy+x^{2}=0$, $y>1$, $x(e)=e$. Then $x(e^{2})$ is equal to :
  14. Let $f$ be a twice differentiable function defined on $\mathbf{R}$ such that $f(0)=1$, $f'(0)=2$ and $f'(x)\neq0$ for all $x\in\mathbf{R}$. If …
  15. Let $y=y(x)$ be the solution of the differential equation …
  16. Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y$; $y(0)=0$ :
  17. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2x(x+y)^{3}-x(x+y)-1$, $y(0)=1$. Then …
  18. Let $y=y(x)$ be the solution curve of the differential equation $\sec y\frac{dy}{dx}+2x\sin y=x^{3}\cos y$, $y(1)=0$. Then $y(\sqrt{3})$ is equal to :
  19. Let $x=x(y)$ be the solution of the differential equation $2(y+2)\log_e(y+2)\,dx+\left(x+4-2\log_e(y+2)\right)dy=0$, $y>-1$ with $x\left(e^{4}-2\right)=1$. Then …
  20. If $x=x(y)$ is the solution of the differential equation $2\sqrt{\sin x+\tan^{-1}y}\cdot\frac{dy}{dx}=\left(\cos x+\frac{1}{1+y^{2}}\frac{dy}{dx}\right)$ satisfying …
  21. Let $y=y_1(x)$ and $y=y_2(x)$ be the solution curves of the differential equation $\frac{dy}{dx}=y+7$ with initial conditions $y_1(0)=0$ and $y_2(0)=1$ respectively. …
  22. The solution curve of the differential equation $y\frac{dx}{dy}=x(\log_ex-\log_ey+1)$, $x>0$, $y>0$ passing through the point $(e,1)$ is
  23. If the solution curve of the differential equation $\left(2x-10y^{3}\right)dy+y\,dx=0$, passes through the points $(0,1)$ and $(2,\beta)$, then $\beta$ is a root of the …
  24. The slope of tangent at any point $(x,y)$ on a curve $y=y(x)$ is $\frac{x^{2}+y^{2}}{2xy}$, $x>0$. If $y(2)=0$, then a value of $y(8)$ is
  25. Let $y(x)$ be the solution of the differential equation $2x^{2}dy+(e^{y}-2x)dx=0$, $x>0$. If $y(e)=1$, then $y(1)$ is equal to :
  26. Let us consider a curve, $y=f(x)$ passing through the point $(-2,2)$ and the slope of the tangent to the curve at any point $(x,f(x))$ is given by $f(x)+xf'(x)=x^{2}$. …
  27. Let $\int_{0}^{x}\sqrt{1-\left(y'(t)\right)^{2}}\,dt=\int_{0}^{x}y(t)\,dt$, $0\le x\le3$, $y\ge0$, $y(0)=0$. Then at $x=2$, $y''+y+1$ is equal to
  28. Let $y=y(x)$ be the solution of the differential equation $\operatorname{cosec}^{2}x\,dy+2\,dx=(1+y\cos2x)\operatorname{cosec}^{2}x\,dx$, with …
  29. A curve passes through the point $(-1,4)$. If the normal to it at any point $(x,y)$ on it passes through $(2,0)$, then its equation is
  30. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+3\left(\tan^{2}x\right)y+3y=\sec^{2}x$, $y(0)=\frac{1}{3}+e^{3}$. Then …
  31. If $y=y(x)$ is the solution of the differential equation, $\sqrt{1-e^{2x}}\,dy-\sqrt{4-y^{2}}\cdot e^{x}dx=0$ such that $y(0)=1$, then $y(-\log_e2)$ is equal to
  32. Let $y=y(x)$ be a solution curve of the differential equation $\left(1-x^{2}y^{2}\right)dx=y\,dx+x\,dy$. If the line $x=1$ intersects the curve $y=y(x)$ at $y=2$ and the …
  33. If the solution curve $f(x,y)=0$ of the differential equation $(1+\log_ex)\frac{dx}{dy}-x\log_ex=e^{y}$, $x>0$, passes through the points $(1,0)$ and $(\alpha,2)$, then …
  34. Let $y=y(x)$, $y>0$, be a solution curve of the differential equation $\left(1+x^{2}\right)dy=y(x-y)dx$. If $y(0)=1$ and $y(2\sqrt{2})=\beta$, then
  35. Let $y=y(x)$ be the solution of the differential equation $\sec x\,dy+\{2(1-x)\tan x+x(2-x)\}dx=0$ such that $y(0)=2$. Then $y(2)$ is equal to :
  36. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{5}{x\left(x^{5}+1\right)}y=\frac{\left(x^{5}+1\right)^{2}}{x^{7}}$, $x>0$. If $y(1)=2$ …
  37. The temperature $T(t)$ of a body at time $t=0$ is $160^{\circ}$F and it decreases continuously as per the differential equation $\frac{dT}{dt}=-K(T-80)$, where $K$ is a …
  38. Let $f$ be a differentiable function such that $x^{2}f(x)-x=4\int_{0}^{x}tf(t)\,dt$, $f(1)=\frac{2}{3}$. Then $18f(3)$ is equal to
  39. Let $y=y(x)$ be the solution curve of the differential equation $x\left(x^{2}+e^{x}\right)dy+\left(e^{x}(x-2)y-x^{3}\right)dx=0$, $x>0$, passing through the point …
  40. Let $f:[1,\infty)\to[2,\infty)$ be a differentiable function. If $10\int_{1}^{x}f(t)\,dt=5xf(x)-x^{5}-9$ for all $x\ge1$, then the value of $f(3)$ is :
  41. The differential equation satisfied by the system of parabolas $y^{2}=4a(x+a)$ is :
  42. If $\frac{dy}{dx}=\frac{2^{x}y+2^{y}\cdot2^{x}}{2^{x}+2^{x+y}\log_e2}$, $y(0)=0$, then for $y=1$, the value of $x$ lies in the interval :
  43. Let $y=y(x)$ be the solution of the differential equation $x\,dy=(y+x^{3}\cos x)dx$ with $y(\pi)=0$, then $y\left(\frac{\pi}{2}\right)$ is equal to :
  44. Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{dx}{dt}+ax=0$ and $\frac{dy}{dt}+by=0$ respectively, $a,b\in\mathbf{R}$. Given that $x(0)=2$ …
  45. Let $f:\mathbf{R}\to\mathbf{R}$ be a twice differentiable function such that $(\sin x\cos y)\left(f(2x+2y)-f(2x-2y)\right)=(\cos x\sin y)\left(f(2x+2y)+f(2x-2y)\right)$ …
  46. Let $y=y(x)$ be the solution of the differential equation $\left(x^{2}+1\right)y'-2xy=\left(x^{4}+2x^{2}+1\right)\cos x$, $y(0)=1$. Then $\int_{-3}^{3}y(x)\,dx$ is :
  47. Let $f(x)$ be a positive function such that the area bounded by $y=f(x)$, $y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4a^{2}+a-1$. Then the differential equation, whose …
  48. The solution of the differential equation $\left(x^{2}+y^{2}\right)dx-5xy\,dy=0$, $y(1)=0$, is :
  49. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)\left((y+1)e^{x^{2}/2}-x\right)$, $0<x<2.1$, with $y(2)=0$. Then the value of …
  50. Let $y=y(x)$ be the solution of the differential equation $\left(1+x^{2}\right)\frac{dy}{dx}+y=e^{\tan^{-1}x}$, $y(1)=0$. Then $y(0)$ is

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