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
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