Questions from Differential Equations

 Q1. For the diffrential equation $\frac{dy}{dt}+5y=0$ with $y(0)=1,$ the general solution is

    a) $e^{5t}$         b) $e^{-5t}$        c) $5e^{-5t}$            d) $e^{\sqrt{-5t}}$

Q2. $y=e^{-2x}$ is a solution of the differential equation $y''+y'-2y=0$

a) True          b) False

Q3.  The differetial equation $\frac{dy}{dx}+Py=Q$, is a linear equation of first order only if

a) P is a constant But Q is a function of y

b) P and Q are functions of y (or) constants.

c) P is a function of y but Q is a constant.

d) P and Q are function of x (or) constant.

Q4. If c is a constant, then the solution of $\frac{dy}{dx}=1+y^2$ is

a)$y=sin(x+c)$          b) $y=cos (x+c)$         c) $y=tan(x+c)$          d) $y=e^x+c$

Q5. The solution of the differential equation $\frac{dy}{dx}+y^2=0$ is

a) $y=\frac{1}{x+c}$         b) $y=-\frac{x^3}{3}+c$ c) $y=ce^x$ d) Unsolvable as equation is non-liear

Q6. Biotransformation of an organic compound having concentration (x) can be modeled using an ordinary differential equation $\frac{dx}{dt}+kx^2=0$, where $k$ is the reaction rate constant. If $x=-a$ at $t=0$ then solution of the equation is

a)$x=ae^{-kt}$         b)$\frac{1}{x}=\frac{1}{a}+kt$         c) $x=a(1-e^{-kt})$         d)$x=a+kt$

 Q7. The differential equation $\left[1+\left(\frac{dy}{dx}\right)^2\right]^3=C^2\left[\frac{d^2y}{dx^2}\right]^2$

a) 2nd order and 3rd degree  b) 3rd order 2nd degree 

c) 2nd order 2nd degree  d) 3rd order 3rd degree.

Q8. The solution of the first order differential equation $\frac{dx}{dt}=-3x,~x(0)=x_0$ is

a) $x(t)=x_0e^{-3t}$      b) $x(t)=x_0e^{3t}$      c)$x(t)=x_0e^{-t/3}$       d) $x(t)=x_0e^{-t}$

Q9. Transformation to linear form by substitutiong $v=y^{1-n}$ of the equation $\frac{dy}{dt}+p(t)y=q(t)y^n, n>0$ will be

a) $\frac{dv}{dt}+(1-n)pv=(1-n)q$

b) $\frac{dv}{dt}+(1+n)pv=(1+n)q$

c) $\frac{dv}{dt}+(1+n)pv=(1-n)q$

d) $\frac{dv}{dt}+(1+n)pv=(1-n)q$

Q10. If $x^2\left(\frac{dy}{dx}\right)+2xy= \frac{2ln x}{x}$ and $y(1)=0$ then what is $y(e)$?

a) $e$          b) 1          c) $\frac{1}{e}$          d) $\frac{1}{e^2}$

Q11. The solution of the differential equation $x^2\frac{dy}{dx}+2xy-x+1=0$ given that at $x=1, y=0$ is

a) $\frac{1}{2}-\frac{1}{x}+\frac{1}{2x^2}$ 

        b) $\frac{1}{2}-\frac{1}{x}-\frac{1}{2x^2}$

         c) $\frac{1}{2}+\frac{1}{x}+\frac{1}{2x^2}$ 

        d) -$\frac{1}{2}+\frac{1}{x}+\frac{1}{2x^2}$

Q12. The solution of the differential equation $\frac{dy}{dx}+2xy=e^{-x^2}$ with $y(0)=1$ is

a) $(1+x)e^{x^2}$     b)$(1+x)e^{-x^2}$        c) $(1-x)e^{x^2}$        d)$(1+x)e^{-x^2}$

Q13. The solution for the differential equation $\frac{dy}{dx}=x^2y$ with the condition that $y=1$ at $x=0$ is

a) $y=e^{\frac{1}{2x}}$         b) $ln(y)=\frac{x^3}{3}+4$          c) $ln(y)=\frac{x^2}{2}$         d) $y=e^{\frac{x^3}{3}}$

Q14. The solution for the differential equation $\frac{dy}{dx}=y^2$ with initial value $y(0)=1$ is bounded in the internal is

a)$-\infty \leq x\leq \infty$    b) $-\infty \leq x\leq 1$     c) $x<1, x>1$         d) $-2 \leq x\leq 2$

Q15. Consider the differential equation $\frac{dy}{dx}=1+y^2$. Which one of the following can be particular solution of this differential equation?

a)$y=tan(x+3)$            b)$y=tanx +3$        c)$x=tan(y+3)$         d)$x=tany+3$

Q16. Which of the following is a solution to the differential equation $\frac{d}{dt}x(t)+3x(t)=0, x(0)=2?$

a)$x(t)=3e^{-t}$     b)$x(t)=2e^{-3t}$       c)$x(t)=\frac{-3}{2}t^2$       d) $x(t)=3t^2$

Q16. Solution of the differential equation $3y\frac{dy}{dx}+2x=0$ represents a family of \\

a) Ellipse         b) Circles         c)Parabolas     d) Hyperbolas.

Q17. The order of the differential equation $\frac{d^2y}{dt^2}+\left(\frac{dy}{dt}\right)^3+y^4=e^{-t}$ is

a) 1         b) 2             c)3            d)4

Q18.The solution of $x\frac{dy}{dx}+y=x^4$ with condition $y(1)=\frac{6}{5}$.

Q19. The solution of the differential equation $\frac{dy}{dx}-y^2=1$ satisfying the condition $y(0)=1$ is.

Q20.Which one of the following differential equations has a solution given by the function $y=5 sin\left(3x+\frac{\pi}{3}\right)$.

Q21. The order and degree of a differential equation $\frac{d^3y}{dx^3}+4\sqrt{\left(\frac{dy}{dx}\right)^3+y^2}=0$ are respectively.\\

a) 3 and 2         b) 2 and 3              c)3 and 3              d) 3 and 1

Q22.  Consider the differential equation $\frac{dy}{dx}+y=e^x$ with $y(0)=1$. Then find the value of $y(1)$ is

Q23.  With K as constant, the possible solution for the first order differential equation $\frac{dy}{dx}=3e^{-3x}$ is

Q24. The solution of the differential equation $\frac{dy}{dx}=ky, ~y(0)=c$ is

Q25. Consider the differential equation $\frac{dy}{dx}=(1+y^2)x$. The general solution with constant C is

Q26. The solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=x$ with the condition that $y=1$ at $x=1$ is

Q27. The integrating factor for the differential equation $\frac{dP}{dt}+k_2P=k_1L_0e^{-k_1t}$ is

Q28. Which one of the following is a linear non-homogeneous differential equation, where $x$ and $y$ are independent and dependent variable respectively?

a) $\frac{dy}{dx}+xy=e^{-x}$         b)$\frac{dy}{dx}+xy=0$         c)$\frac{dy}{dx}+xy=e^{-y}$         d) $\frac{dy}{dx}+e^{-y}=0$

Q29.  The solution of the initial value problem $\frac{dy}{dx}=-2xy; y(0)=2$ is

Q30. The general solution of the differential equation $\frac{dy}{dx}=cos(x+y),$ with c as a constant, is

Q31. The general solution of the differential equation $\frac{dy}{dx}=\frac{1+cos 2y}{1-cos 2x}$

Q32. Consider the differential equation $\frac{dx}{dt}=10-0.2x$ with initial condition $x(0)=1$. The response $x(t)$ for $t>0$ is

Q33. Consider the following differential equation $\frac{dy}{dt}=-5y$ initial condition: $y=2$ at $t=3$.

Q34. Consider the following differential equation $x(ydx+xdy)cos \frac{y}{x}=y(xdy-ydx)sin\frac{y}{x}$ which of the following is the solution of the above equation.

Q35. The solution of the equation $\frac{dQ}{dt}+Q=1$ with $Q=0~at~t=0$ is

Q36. A curve passes through the point $(x=1, y=0)$ and satisfies the differential equation $\frac{dy}{dx}=\frac{x^2+y^2}{2y}+\frac{y}{x}.$ The equation that describes the curve is 

a)$ln\left(1+\frac{y^2}{x^2}\right)=x-1$

b)$\frac{1}{2}ln\left(1+\frac{y^2}{x^2}\right)=x-1$

c)$ln\left(1+\frac{y}{x}\right)=x-1$

d)$\frac{1}{2}ln\left(1+\frac{y^2}{x^2}\right)=x-1$

Q37. The solution of the equation $x\frac{dy}{dx}+y=0$ passing through the point $(1, 1)$.

Q38. If $y$ is the solution of the differential equation $y^3\frac{dy}{dx}+x^3=0, y(0)=1$ The value of $y(-1)$ is

a) -2         b) -1         c)0        d)1

Q39. For the equation $\frac{dy}{dx}+7x^2y=0$, if $y(0)=3/7$ then the value of $y(1)$ 

Q40  The differential equation $\frac{dy}{dx}+4y=5$ is valid in the domain $0\leq x\leq 1$ with $y(0)=2.25$ The solution of the differential equation is.

Q41. The family of curves represented by the solution of the equation $\frac{dy}{dx}=-\left(\frac{x}{y}\right)^n$ For $n=-1$ and $n=+1$ respectively, are

a) Hyperbola and Parabolas b) Circles and Hyperbolas

c)Parabolas and Circles  d) Hyperbolas and circles.

Q42. What is the solution of the differential equation $\frac{dy}{dx}=\frac{x}{y}$, with the initial condition, at $x=0, y=1$?

Q43.  Find the solution of the given differential equation $\left(\frac{dy}{dx}\right)xln x=y$

Q44. The solution of the ordinary differential equation $\frac{dy}{dx}+3y=1$ Subject to the initial condition $y=1$ at $x=0$ is

Q45. One of the points which lies on the solution curves of the following differential equation $2xydx+(x^2+y^2)dy=0$ with the initial condition $y(1)=1$ is

a) (1,1)     b)(0,0)         c)(0,1)     d)(2,1).

Q46. Obtain the differential equation of all circles each of which touches the axis of $x$ at the origin.

Q47. Solve the differential equations given below:

A. $(y^2e^{xy^2}+4x^3)dx+(2xye^{xy^2}-3y^2)dy=0$.

B. $(x^3-3x^2y+2xy^2)dx-(x^3-2x^2y+y^3)dy=0$

C. $(xy^2-e^{\frac{1}{x^3}})dx-x^2ydy=0$.

D. $2ydx-xdy=xy^3dy$

E. $\frac{dy}{dx}+2xy=x^2+y^2$

F. $x^2dy+y(x+y)dx=0$

Important Links:

1) AKU previous year questions and solution from Matrices