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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 251-300 of 313

  1. If $y=y(x)$ and $\dfrac{2+\sin x}{y+1}\left(\dfrac{dy}{dx}\right)=-\cos x$, $y(0)=1$, then $y\left(\dfrac{\pi}{2}\right)$ equals
  2. The order of the differential equation whose general solution is given by $y=(C_1+C_2)\cos(x+C_3)-C_4e^{x+C_5}$, where $C_1,C_2,C_3,C_4,C_5$ are arbitrary constants, is
  3. A solution of the differential equation $\left(\dfrac{dy}{dx}\right)^{2}-x\dfrac{dy}{dx}+y=0$ is
  4. The differential equation representing the family of curves $y^{2}=2c\left(x+\sqrt{c}\right)$, where $c$ is a positive parameter, is of
  5. For the primitive integral equation $y\,dx+y^{2}\,dy=x\,dy$; $x\in R$, $y>0$, $y=y(x)$, $y(1)=1$, then $y(-3)$ is
  6. Let $f$ be a twice differentiable non-negative function such that $(f(x))^2=25+\int_0^x\left((f(t))^2+(f'(t))^2\right)dt$. Then the mean of …
  7. If the solution curve $y=f(x)$ of the differential equation $(x^2-4)y'-2xy+2x(4-x^2)^2=0$, $x>2$, passes through the point $(3,15)$, then the local maximum value of $f$ …
  8. If the curve $y=f(x)$ passes through the point $(1,e)$ and satisfies the differential equation $dy=y(2+\log_{e}x)\,dx$, $x>0$, then $f(e)$ is equal to :
  9. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(1+x+x^{2})(1-y+y^{2})$, $y(0)=\frac12$. Then $(2y(1)-1)$ is equal to
  10. For any real numbers $\alpha$ and $\beta$, let $y_{\alpha,\beta}(x)$, $x\in\mathbb{R}$, be the solution of the differential equation $$\frac{dy}{dx}+\alpha y = x …
  11. Let the solution curve of the differential equation $x\,dy-y\,dx=\sqrt{x^2+y^2}\,dx$, $x>0$, $y(1)=0$, be $y=y(x)$. Then $y(3)$ is equal to :
  12. Let $f:[1,\infty)\to\mathbf{R}$ be a differentiable function. If $6\int_1^x f(t)\,dt=3xf(x)+x^3-4$ for all $x\ge1$, then the value of $f(2)-f(3)$ is :
  13. If $y=y(x)$ satisfies the differential equation $16\left(\sqrt{x\sqrt x+9x}\right)\left(4+\sqrt{9+\sqrt x}\right)\cos y\,dy=(1+2\sin y)\,dx$, $x>0$ and …
  14. Let $y=y(x)$ be the solution curve of the differential equation $(1+x^2)\,dy+(y-\tan^{-1}x)\,dx=0$, with initial condition $y(0)=1$. Then find the value of $y(1)$.
  15. Let $y=y(x)$ be the solution of the differential equation $\sec x\frac{dy}{dx}-2y=2+3\sin x$, $x\in\left(-\frac\pi2,\frac\pi2\right)$, $y(0)=-\frac74$. Then …
  16. Let $y=y(x)$ be the solution of the differential equation $x^4\,dy+\left(4x^3y+2\sin x\right)dx=0$, $x>0$, $y\left(\frac\pi2\right)=0$. Then …
  17. Let $y=y(x)$ be a differentiable function in the interval $(0,\infty)$ such that $y(1)=2$, and $\lim_{t\to x}\left(\frac{t^2y(x)-x^2y(t)}{x-t}\right)=3$ for each $x>0$. …
  18. Let $y=y(x)$ be the solution of the differential equation $x\frac{dy}{dx}-y=x^2\cot x$, $x\in(0,\pi)$. If $y\left(\frac\pi2\right)=\frac\pi2$, then …
  19. Let $y=y(x)$ be the solution of the differential equation $x\frac{dy}{dx}-\sin2y=x^3\left(2-x^3\right)\cos^2y$, $x\ne0$. If $y(2)=0$, then $\tan(y(1))$ is equal to :
  20. The differential equation $\dfrac{dy}{dx}=\dfrac{\sqrt{1-y^{2}}}{y}$ determines a family of circles with
  21. Let $f:[1,\infty)\to[2,\infty)$ be a differentiable function such that $f(1)=2$. If $$6\int_1^x f(t)\,dt=3xf(x)-x^3$$ for all $x\ge1$, then the value of $f(2)$ is
  22. The function $y=f(x)$ is the solution of the differential equation $$\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$$ in $(-1,1)$ satisfying $f(0)=0$. Then …
  23. If $x\frac{dy}{dx}=y(\log y-\log x+1)$, then the solution of the equation is :
  24. Let f be a real-valued differentiable function on $\mathbf R$ (the set of all real numbers) such that $f(1)=1$. If the y-intercept of the tangent at any point P(x, y) on …
  25. Let $y'(x)+y(x)g'(x)=g(x)g'(x)$, $y(0)=0$, $x\in\mathbb{R}$, where $f'(x)$ denotes $\frac{df(x)}{dx}$ and $g(x)$ is a given non-constant differentiable function on …
  26. A curve passes through the point $\left(1,\frac{\pi}{6}\right)$. Let the slope of the curve at each point $(x,y)$ be $\frac{y}{x}+\sec\left(\frac{y}{x}\right)$, $x>0$. …
  27. Let $y(x)$ be a solution of the differential equation $(1+e^x)y'+ye^x=1$. If $y(0)=2$, then which of the following statements is (are) true?
  28. Consider the family of all circles whose centers lie on the straight line $y=x$. If this family of circles is represented by the differential equation $Py''+Qy'+1=0$ …
  29. Let $f:(0,\infty)\to\mathbb{R}$ be a differentiable function such that $f'(x)=2-\frac{f(x)}{x}$ for all $x\in(0,\infty)$ and $f(1)\neq1$. Then
  30. If $y = y(x)$ satisfies the differential equation $$8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right)dy = \left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)^{-1}dx, \quad x>0$$ and …
  31. Let $f:[0,\infty)\to\mathbb{R}$ be a continuous function such that $$f(x)=1-2x+\int_0^x e^{x-t}f(t)\,dt$$ for all $x\in[0,\infty)$. Then, which of the following …
  32. Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function with $f(0)=1$ and satisfying the equation $$f(x+y)=f(x)f'(y)+f'(x)f(y)\text{ for all }x,y\in\mathbb{R}.$$ …
  33. Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation $$\frac{dy}{dx}=(2+5y)(5y-2),$$ then the …
  34. Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y\frac{dx}{dy}\right)\sin\left(\frac{x}{y}\right)$, $y>0$ and $x(1)=\frac{\pi}{2}$. Then …
  35. If $\frac{dy}{dx}=y+3>0$ and $y(0)=2$, then $y(\ln2)$ is equal to :
  36. The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{dp(t)}{dt}=0.5\,p(t)-450$. If $p(0)=850$, then the time at which …
  37. Let $u(x)$ and $v(x)$ satisfy the differential equation $\dfrac{du}{dx}+p(x)u=f(x)$ and $\dfrac{dv}{dx}+p(x)v=g(x)$, where $p(x)$, $f(x)$ and $g(x)$ are continuous …
  38. If $\dfrac{dw(t)}{dt}=t\sqrt{121-t^{2}}$ and $w(0)=0$, then the value of $3w\left(\sqrt{21}\right)$ is $\_\_\_\_\_\_\_\_$ .
  39. If $y=y(x)$ is the solution of the differential equation, $\dfrac{dy}{dx}+2y\tan x=\sin x$, $y\left(\dfrac{\pi}{3}\right)=0$, then the maximum value of the function …
  40. Let the curve $y=y(x)$ be the solution of the differential equation, $\dfrac{dy}{dx}=2(x+1)$. If the numerical value of area bounded by the curve $y=y(x)$ and $x$-axis …
  41. If the solution $y=y(x)$ of the differential equation $\left(x^{4}+2x^{3}+3x^{2}+2x+2\right)dy-\left(2x^{2}+2x+3\right)dx=0$ satisfies $y(-1)=-\dfrac{\pi}{4}$, then …
  42. Let the solution $y=y(x)$ of the differential equation $\dfrac{dy}{dx}-y=1+4\sin x$ satisfy $y(\pi)=1$. Then $y\left(\dfrac{\pi}{2}\right)+10$ is equal to …
  43. Let $y=y(x)$ be the solution of the differential equation $(x+y+2)^{2}dx=dy$, $y(0)=-2$. Let the maximum and minimum values of the function $y=y(x)$ in …
  44. Let $y=y(x)$ be the solution of the differential equation $\left(x^{2}+4\right)^{2}dy+\left(2x^{3}y+8xy-2\right)dx=0$. If $y(0)=0$, then $y(2)$ is equal to
  45. If $y=y(x)$ is the solution of the differential equation $\dfrac{dy}{dx}+2y=\sin(2x)$, $y(0)=\dfrac{3}{4}$, then $y\left(\dfrac{\pi}{8}\right)$ is equal to :
  46. The differential equation of the family of circles passing through the origin and having centre at the line $y=x$ is :
  47. Let $y=y(x)$ be the solution of the differential equation $\cos x\dfrac{dy}{dx}+2y\sin x=\sin2x$, $x\in\left(0,\dfrac{\pi}{2}\right)$. If $y(\pi/3)=0$, then $y(\pi/4)$ …
  48. The general solution of the differential equation $\sqrt{1+x^{2}+y^{2}+x^{2}y^{2}}+xy\dfrac{dy}{dx}=0$ is : (where $C$ is a constant of integration)
  49. Let $x=x(y)$ be the solution of the differential equation $y^{2}dx+\left(x-\dfrac{1}{y}\right)dy=0$. If $x(1)=1$, then $x\left(\dfrac{1}{2}\right)$ is :
  50. If $x=f(y)$ is the solution of the differential equation $\left(1+y^{2}\right)+\left(x-2e^{\tan^{-1}y}\right)\dfrac{dy}{dx}=0$ …

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