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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 151-200 of 313

  1. Let $y=y(x)$ be solution of the following differential equation $e^{y}\frac{dy}{dx}-2e^{y}\sin x+\sin x\cos^{2}x=0$, $y\left(\frac{\pi}{2}\right)=0$. If …
  2. If $\frac{dx}{dy}=\frac{1+x-y^{2}}{y}$, $x(1)=1$, then $5x(2)$ is equal to $\_\_\_\_\_\_$ .
  3. Let $y=y(x)$ be the solution of the differential equation $dy=e^{\alpha x+y}dx$; $\alpha\in\mathbf{N}$. If $y(\log_e2)=\log_e2$ and $y(0)=\log_e\left(\frac12\right)$ …
  4. The differential equation representing the family of curves $y^{2}=2c\left(x+\sqrt c\right)$, where $c>0$, is a parameter, is of order and degree as follows :
  5. Solution of the differential equation $y\,dx+\left(x+x^{2}y\right)dy=0$ is
  6. The solution of the differential equation $\frac{dy}{dx}+\frac y2\sec x=\frac{\tan x}{2y}$, where $0\le x<\frac\pi2$ and $y(0)=1$, is given by
  7. If the general solution of the differential equation $y'=\frac yx+\Phi\left(\frac xy\right)$, for some function $\Phi$, is given by $y\ln|cx|=x$, where $c$ is an …
  8. For $x\in R,\ x\neq0$, if $y(x)$ is a differentiable function such that $x\int_{1}^{x}y(t)\,dt=(x+1)\int_{1}^{x}t\,y(t)\,dt$, then $y(x)$ equals (where $C$ is a constant)
  9. If $y(x)$ is the solution of the differential equation $(x+2)\frac{dy}{dx}=x^2+4x-9$, $x\ne-2$ and $y(0)=0$, then $y(-4)$ is equal to
  10. The solution of the differential equation $\frac{dy}{dx}=\frac{x+y}{x}$ satisfying the condition $y(1)=1$ is :
  11. If $y=y(x)$ is the solution curve of the differential equation $x^{2}dy+\left(y-\frac1x\right)dx=0$; $x>0$ and $y(1)=1$, then $y\left(\frac12\right)$ is equal to :
  12. Solution of the differential equation $\cos x\,dy=y(\sin x-y)\,dx$, $0<x<\frac\pi2$ is
  13. The degree and order of the differential equation of the family of all parabolas whose axis is $x$-axis, are respectively.
  14. Let $y=y(x)$ be the solution curve of the differential equation, $\left(y^2-x\right)\frac{dy}{dx}=1$, satisfying $y(0)=1$. This curve intersects the $X$-axis at a point …
  15. The equation of the curve passing through the origin and satisfying the differential equation $\left(1+x^{2}\right)\frac{dy}{dx}+2xy=4x^{2}$ is :
  16. The solution of the differential equation $y\,dx-\left(x+2y^{2}\right)dy=0$ is $x=f(y)$. If $f(-1)=1$, then $f(1)$ is equal to
  17. If the differential equation representing the family of all circles touching $x$-axis at the origin is $\left(x^{2}-y^{2}\right)\frac{dy}{dx}=g(x)y$, then $g(x)$ equals
  18. Let $y=y(x)$ be the solution of the differential equation $\sin x\frac{dy}{dx}+y\cos x=4x$, $x\in(0,\pi)$. If $y\left(\frac\pi2\right)=0$, then $y\left(\frac\pi6\right)$ …
  19. Let $y=y(x)$ be the solution of the differential equation $x\tan\left(\frac yx\right)dy=\left(y\tan\left(\frac yx\right)-x\right)dx$, $-1\le x\le1$ …
  20. The solution of the differential equation $\left(1+y^{2}\right)+\left(x-e^{\tan^{-1}y}\right)\frac{dy}{dx}=0$, is :
  21. If $y(x)$ is the solution of the differential equation $\frac{dy}{dx}+\left(\frac{2x+1}{x}\right)y=e^{-2x}$, $x>0$, where $y(1)=\frac12e^{-2}$, then :
  22. Let $I$ be the purchase value of an equipment and $V(t)$ be the value after it has been used for $t$ years. The value $V(t)$ depreciates at a rate given by differential …
  23. The differential equation whose solution is $Ax^{2}+By^{2}=1$ where $A$ and $B$ are arbitrary constants is of
  24. Let $y=y(t)$ be a solution of the differential equation $\frac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$ where $\alpha>0$, $\beta>0$ and $\gamma>0$. Then …
  25. If $\frac{dy}{dx}+y\tan x=\sin2x$ and $y(0)=1$, then $y(\pi)$ is equal to
  26. The order and degree of the differential equation $\left(1+3\frac{dy}{dx}\right)^{2/3}=4\frac{d^{3}y}{dx^{3}}$ are
  27. Consider the differential equation : $\frac{dy}{dx}=\frac{y^{3}}{2\left(xy^{2}-x^{2}\right)}$ Statement-1: The substitution $z=y^{2}$ transforms the above equation into …
  28. The solution of the equation $\frac{d^{2}y}{dx^{2}}=e^{-2x}$
  29. Let $y=y(x)$ be the solution of the differential equation $e^{x}\sqrt{1-y^{2}}dx+\left(\frac yx\right)dy=0$, $y(1)=-1$. Then the value of $(y(3))^{2}$ is equal to :
  30. If $(2+\sin x)\frac{dy}{dx}+(y+1)\cos x=0$ and $y(0)=1$, then $y\left(\frac\pi2\right)$ is equal to :
  31. If a curve passes through the point $\left(2,\frac72\right)$ and has slope $\left(1-\frac{1}{x^{2}}\right)$ at any point $(x,y)$ on it, then the ordinate of the point on …
  32. If $y=y(x)$ is the solution of the differential equation $e^{y}\left(\frac{dy}{dx}-1\right)=e^{x}$ such that $y(0)=0$, then $y(1)$ is equal to
  33. A tangent to the curve $y=f(x)$ at $P(x,y)$ meets $x$-axis at $A$ and $y$-axis at $B$. If $AP:BP=1:3$ and $f(1)=1$, then the curve also passes through the point
  34. Let $y=f(x)$ be the solution of the differential equation $y(x+1)\,dx-x^{2}dy=0$, $y(1)=e$. Then $\lim_{x\to0^{+}}f(x)$ is equal to
  35. If $y=y(x)$ is the solution curve of the differential equation $\left(x^2-4\right)dy-\left(y^2-3y\right)dx=0$, $x>2$, $y(4)=\frac32$ and the slope of the curve is never …
  36. Let $y=y(x)$ be the solution of the differential equation $x\log_ex\frac{dy}{dx}+y=x^{2}\log_ex$, $(x>1)$. If $y(2)=2$, then $y(e)$ is equal to
  37. If $\sin\left(\frac yx\right)=\log_e|x|+\frac\alpha2$ is the solution of the differential equation $x\cos\left(\frac yx\right)\frac{dy}{dx}=y\cos\left(\frac yx\right)+x$ …
  38. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+2y=f(x)$, where $f(x)=\begin{cases}1,&x\in[0,1]\\0,&\text{otherwise}\end{cases}$. If $y(0)=0$ …
  39. Let $y=y(x)$ be the solution of the differential equation $\left(x^{2}+1\right)^{2}\frac{dy}{dx}+2x\left(x^{2}+1\right)y=1$ such that $y(0)=0$. If …
  40. If a curve $y=f(x)$ passes through the point $(1,-1)$ and satisfies the differential equation $y(1+xy)\,dx=x\,dy$, then $f\left(-\frac12\right)$ is equal to
  41. The curve satisfying the differential equation $\left(x^{2}-y^{2}\right)dx+2xy\,dy=0$ and passing through the point $(1,1)$ is
  42. The solution of the differential equation, $\frac{dy}{dx}=(x-y)^{2}$, when $y(1)=1$, is :
  43. If $y=y(x)$ is the solution of the differential equation, $x\frac{dy}{dx}+2y=x^2$ satisfying $y(1)=1$, then $y\left(\frac12\right)$ is equal to
  44. Consider the differential equation, $y^2dx+\left(x-\frac1y\right)dy=0$. If value of $y$ is $1$ when $x=1$, then the value of $x$ for which $y=2$, is
  45. A curve amongst the family of curves represented by the differential equation, $\left(x^2-y^2\right)dx+2xy\,dy=0$ which passes through $(1,1)$, is :
  46. The general solution of the differential equation $\sin2x\left(\frac{dy}{dx}-\sqrt{\tan x}\right)-y=0$ is :
  47. The curve satisfying the differential equation $y\,dx-\left(x+3y^{2}\right)dy=0$ and passing through the point $(1,1)$ also passes through the point :
  48. Let $f:[0,1]\to\mathbb R$ be such that $f(xy)=f(x)\cdot f(y)$, for all $x,y\in[0,1]$, and $f(0)\neq0$. If $y=y(x)$ satisfies the differential equation …
  49. The solution of the differential equation $x\frac{dy}{dx}+2y=x^{2}$, $(x\neq0)$ with $y(1)=1$, is
  50. If $\frac{dy}{dx}+\frac{3}{\cos^{2}x}y=\frac{1}{\cos^{2}x},\ x\in\left(-\frac\pi3,\frac\pi3\right)$, and $y\left(\frac\pi4\right)=\frac43$, then …

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