Ordinary Differential Equations – Mathematics Optional PYQs (2008–2026)

Here are all 118 Ordinary Differential Equations questions asked in UPSC Mathematics Optional Paper I from 2008 to 2026, arranged under the 5 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

4.1] First order first degree equations

42 questions

1. If $y_1$ and $y_2$ are two solutions of $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$, and $y_2=y_{1}u$, then show that $u=1+a{e}^{-\int \left(\frac{Q}{y_1}\right)dx}$, where $a$ is a non-zero constant. [2026/10m/5a]

2. Show that ${(x+y+1)}^{-4}$ is an integrating factor of $(2x-y-1)ydx+(2y-x-1)xdy=0$ Hence solve it. [2026/10m/7c(i)]

3. Form the differential equation of all ellipses whose axes coincide with coordinate axes. [2025/10m/5b]

4. Find the orthogonal trajectories of the family of curves $r=c(sec\theta +tan\theta )$, where $c$ is a parameter. [2024/10m/5a]

5. State uniqueness theorem for the existence of unique solution of the initial value problem $\frac{dy}{dx}=f(x,y)$, $y\left(x_0\right)=y_0$ in the rectangular region $R:|x-x_0|\le a$, $|y-y_0|\le b$. Test the existence and uniqueness of the solution of the initial value problem $\frac{dy}{dx}=2\sqrt{y}$, $y\left(1\right)=0$, in a suitable rectangle $R$. If more than one solution exist, then find all the solutions. [2024/15m/7a]

6. Obtain the solution of the initial-value problem $\frac{dy}{dx}-2xy=2,y\left(0\right)=1$ in the form $y=e^{x^2}[1+\sqrt{\pi }erf\left(x\right)]$. [2023/10m/5a]

7. Find the solution of the differential equation : $\frac{dy}{dx}=-\frac{2x{y}^3+2}{3x^2{y}^2+8e^{4y}}$ [2023/10m/7a(i)]

8. Show that the general solution of the differential equation $\frac{dy}{dx}+Py=Q$ can be written in the form $y=\frac{Q}{P}-e^{-\int Pdx}\{C+\int e^{\int Pdx}d\left(\frac{Q}{P}\right)\}$, where $P$, $Q$ are non-zero functions of $x$ and $C$, an arbitrary constant. [2022/10m/5a]

9. Show that the orthogonal trajectories of the system of parabolas : $x^2=4a(y+a)$ belong to the same system. [2022/10m/5b]

10. Find the orthogonal trajectories of the family of confocal conics $\frac{x^2}{a^2+\lambda }+\frac{y^2}{b^2+\lambda }=1$; $a>b>0$ are constants and $\lambda$ is a parameter. Show that the given family of curves is self orthogonal. [2021/10m/8a(i)]

11. Solve the following differential equation : $xcos\left(\frac{y}{x}\right)(ydx+xdy)=ysin\left(\frac{y}{x}\right)(xdy-ydx)$ [2020/10m/5a]

12. Find the orthogonal trajectories of the family of circles passing through the points $(0,2)$ and $(0,-2)$. [2020/10m/5b]

13. Solve the differential equation $(2ysinx+3y^{4}sinxcosx)dx-(4y^3{cos}^{2}x+cosx)dy=0$ [2019/10m/5a]

14. Find $\alpha$ and $\beta$ such that $x^{\alpha }{y}^{\beta }$ is an integrating factor of $(4y^2+3xy)dx-(3xy+2x^2)dy=0$ and solve the equation. [2018/12m/7d]

15. Find $f\left(y\right)$ such that $(2x{e}^y+3y^2)dy+(3x^2+f\left(y\right))dx=0$ is exact and hence solve. [2018/12m/8d]

16. Find the differential equation representing all the circles in the $x$–$y$ plane. [2017/10m/5a]

17. Suppose that the streamlines of the fluid flow are given by a family of curves $xy=c$. Find the equipotential lines, that is, the orthogonal trajectories of the family of curves representing the streamlines. [2017/10m/5b]

18. If the growth rate of the population of bacteria at any time $t$ is proportional to the amount present at that time and population doubles in one week, then how much bacterias can be expected after 4 weeks? [2017/8m/6a(ii)]

19. Solve : $\frac{dy}{dx}=\frac{1}{1+x^2}(e^{{tan}^{-1}x}-y)$ [2016/10m/5c]

20. Show that the family of parabolas $y^2=4cx+4c^2$ is self-orthogonal. [2016/10m/5d]

21. Solve : $\{y(1-xtanx)+x^{2}cosx\}dx-xdy=0$ [2016/10m/6a]

22. Solve the differential equation : $xcosx\frac{dy}{dx}+y(xsinx+cosx)=1.$ [2015/10m/5a]

23. Solve the differential equation : $(2x{y}^4{e}^y+2x{y}^3+y)dx+(x^2{y}^4{e}^y-x^2{y}^2-3x)dy=0.$ [2015/10m/5b]

24. Find the constant $a$ so that ${(x+y)}^a$ is the Integrating factor of $(4x^2+2xy+6y)dx+(2x^2+9y+3x)dy=0$ and hence solve the differential equation. [2015/12m/6a]

25. Justify that a differential equation of the form : $[y+xf(x^2+y^2)]dx+[yf(x^2+y^2)-x]dy=0,$ where $f(x^2+y^2)$ is an arbitrary function of $(x^2+y^2)$, is not an exact differential equation and $\frac{1}{x^2+y^2}$ is an integrating factor for it. Hence solve this differential equation for $f(x^2+y^2)={(x^2+y^2)}^2$. [2014/10m/5a]

26. Find the curve for which the part of the tangent cut-off by the axes is bisected at the point of tangency. [2014/10m/5b]

27. Find the sufficient condition for the differential equation $M(x,y)dx+N(x,y)dy=0$ to have an integrating factor as a function of $(x+y)$. What will be the integrating factor in that case ? Hence find the integrating factor for the differential equation $(x^2+xy)dx+(y^2+xy)dy=0,$ and solve it. [2014/15m/8a]

28. $y$ is a function of $x$, such that the differential coefficient $\frac{dy}{dx}$ is equal to $cos(x+y)+sin(x+y)$. Find out a relation between $x$ and $y$, which is free from any derivative/differential. [2013/10m/5a]

29. Obtain the equation of the orthogonal trajectory of the family of curves represented by $r^n=asinn\theta$, $(r,\theta )$ being the plane polar coordinates. [2013/10m/5b]

30. Solve the differential equation $(5x^3+12x^2+6y^2)dx+6xydy=0$. [2013/10m/6a]

31. Solve $\frac{dy}{dx}=\frac{2xy{e}^{{\left(x/y\right)}^2}}{y^2(1+e^{{\left(x/y\right)}^2})+2x^2{e}^{{\left(x/y\right)}^2}}$ [2012/12m/5a]

32. Find the orthogonal trajectories of the family of curves $x^2+y^2=ax$. [2012/12m/5b]

33. Show that the differential equation $(2xylogy)dx+(x^2+y^2\sqrt{y^2+1})dy=0$ is not exact. Find an integrating factor and hence, the solution of the equation. [2012/20m/6a]

34. Obtain the solution of the ordinary differential equation $\frac{dy}{dx}={(4x+y+1)}^2$, if $y\left(0\right)=1$. [2011/10m/5a]

35. Determine the orthogonal trajectory of a family of curves represented by the polar equation $r=a(1-cos\theta ),$ $(r,\theta )$ being the plane polar coordinates of any point. [2011/10m/5b]

36. Consider the differential equation $y\prime =\alpha x,x>0$ where $\alpha$ is a constant. Show that— (i) if $\varphi \left(x\right)$ is any solution and $\psi \left(x\right)=\varphi \left(x\right)e^{-\alpha x}$, then $\psi \left(x\right)$ is a constant; (ii) if $\alpha <0$, then every solution tends to zero as $x\to \infty$. [2010/12m/5a]

37. Show that the differential equation $(3y^2-x)+2y(y^2-3x)y\prime =0$ admits an integrating factor which is a function of $(x+y^2)$. Hence solve the equation. [2010/12m/5b]

38. Verify that $\frac{1}{2}(Mx+Ny)d\left({log}_e\left(xy\right)\right)+\frac{1}{2}(Mx-Ny)d\left({log}_e\left(\frac{x}{y}\right)\right)=Mdx+Ndy$ Hence show that— (i) if the differential equation $Mdx+Ndy=0$ is homogeneous, then $(Mx+Ny)$ is an integrating factor unless $Mx+Ny\equiv 0$; (ii) if the differential equation $Mdx+Ndy=0$ is not exact but is of the form $f_1\left(xy\right)ydx+f_2\left(xy\right)xdy=0$ then ${(Mx-Ny)}^{-1}$ is an integrating factor unless $Mx-Ny\equiv 0$. [2010/20m/6a]

39. Show that the set of solutions of the homogeneous linear differential equation $y\prime +p\left(x\right)y=0$ on an interval $I=[a,b]$ forms a vector subspace $W$ of the real vector space of continuous functions on $I$. What is the dimension of $W$? [2010/20m/7a]

40. Find the differential equation of the family of circles in the $xy$-plane passing through $(-1,1)$ and $(1,1)$. [2009/20m/6a]

41. Solve : $\frac{dy}{dx}=\frac{y^2(x-y)}{3x{y}^2-x^{2}y-4y^3},y\left(0\right)=1$ [2009/20m/6c]

42. Solve the differential equation $ydx+(x+x^3{y}^2)dy=0.$ [2008/12m/5a]

4.2] First order higher degree equations

12 questions

1. Solve $(1-y^2+\frac{y^4}{x^2}){\left(\frac{dy}{dx}\right)}^2-2\frac{y}{x}\frac{dy}{dx}+\frac{y^2}{x^2}=0$. [2025/10m/5a]

2. Find the general solution and singular solution of the differential equation ${(1+\frac{dy}{dx})}^3=\frac{27}{8a}(x+y){(1-\frac{dy}{dx})}^3$. [2025/10m/7c(i)]

3. Reduce the equation $x^2{p}^2+y(2x+y)p+y^2=0$ to Clairaut’s form by the substitution $y=u$ and $xy=v$. Hence solve the equation and show that $y+4x=0$ is a singular solution of the differential equation. [2023/10m/7a(ii)]

4. Find the general and singular solutions of the differential equation : $(x^2-a^2)p^2-2xyp+y^2+a^2=0$, where $p=\frac{dy}{dx}$. Also give the geometric relation between the general and singular solutions. [2022/10m/8a(i)]

5. Find all possible solutions of the differential equation : $y^{2}logy=xy\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2.$ [2021/15m/7b]

6. Find the general and singular solutions of the differential equation $9p^2{(2-y)}^2=4(3-y)$, where $p=\frac{dy}{dx}$. [2020/10m/8a(ii)]

7. Obtain the singular solution of the differential equation ${\left(\frac{dy}{dx}\right)}^2{\left(\frac{y}{x}\right)}^2{cot}^2\alpha -2\left(\frac{dy}{dx}\right)\left(\frac{y}{x}\right)+{\left(\frac{y}{x}\right)}^2{cosec}^2\alpha =1$ Also find the complete primitive of the given differential equation. Give the geometrical interpretations of the complete primitive and singular solution. [2019/15m/8a]

8. Solve : ${\left(\frac{dy}{dx}\right)}^{2}y+2\frac{dy}{dx}x-y=0$ [2018/13m/6a]

9. Consider the differential equation $xy{p}^2-(x^2+y^2-1)p+xy=0$ where $p=\frac{dy}{dx}$. Substituting $u=x^2$ and $v=y^2$ reduce the equation to Clairaut’s form in terms of $u$, $v$ and $p\prime =\frac{dv}{du}$. Hence, or otherwise solve the equation. [2017/10m/6b(i)]

10. Solve the differential equation $x=py-p^2\mathrm{\text{where}}p=\frac{dy}{dx}.$ [2015/13m/7d]

11. Obtain Clairaut’s form of the differential equation $(x\frac{dy}{dx}-y)(y\frac{dy}{dx}+x)=a^2\frac{dy}{dx}.$ Also find its general solution. [2011/15m/6a]

12. Solve the equation $y-2xp+y{p}^2=0,\mathrm{\text{where}}p=\frac{dy}{dx}.$ [2008/15m/6d]

4.3] Higher order linear equations with constant coefficients

13 questions

1. Solve the differential equation : $\frac{d^{3}y}{d{x}^3}-3\frac{d^{2}y}{d{x}^2}+4\frac{dy}{dx}-2y=e^x+cosx.$ [2023/15m/6a]

2. Solve the differential equation : $\frac{d^{2}y}{d{x}^2}+2y=x^2{e}^{3x}+e^{x}cos2x$ [2021/10m/5a]

3. Determine the complete solution of the differential equation $\frac{d^{2}y}{d{x}^2}-4\frac{dy}{dx}+4y=3x^2{e}^{2x}sin2x$ [2019/10m/5b]

4. Solve : $y″-y=x^2{e}^{2x}$ [2018/10m/5a]

5. Solve : $y‴-6y″+12y\prime -8y=12e^{2x}+27e^{-x}$ [2018/10m/5c]

6. Solve : $y″+16y=32sec2x$ [2018/13m/6c]

7. Solve the initial value problem $\begin{array}{cc}\hfill \hfill & \hfill y″-5y\prime +4y=e^{2t}\hfill \\ \hfill \hfill & \hfill y\left(0\right)=\frac{19}{12},y\prime \left(0\right)=\frac{8}{3}\hfill \end{array}$ [2018/13m/7c]

8. Solve the following simultaneous linear differential equations : $(D+1)y=z+e^x$ and $(D+1)z=y+e^x$ where $y$ and $z$ are functions of independent variable $x$ and $D\equiv \frac{d}{dx}$. [2017/8m/6a(i)]

9. Solve the following initial value differential equations : $20y″+4y\prime +y=0$, $y\left(0\right)=3.2$ and $y\prime \left(0\right)=0$. [2017/7m/6b(ii)]

10. Find a particular integral of $\frac{d^{2}y}{d{x}^2}+y=e^{x/2}sin\frac{x\sqrt{3}}{2}$. [2016/10m/5a]

11. Find the general solution of the equation $y‴-y″=12x^2+6x$. [2012/20m/6b]

12. Obtain the general solution of the second order ordinary differential equation $y″-2y\prime +2y=x+e^{x}cosx,$ where dashes denote derivatives w.r.t. $x$. [2011/15m/6b]

13. Use the method of undetermined coefficients to find the particular solution of $y″+y=sinx+(1+x^2)e^x$ and hence find its general solution. [2010/20m/8a]

4.4] Second order equations with variable coefficients

30 questions

1. Solve $x^2\frac{d^{2}y}{d{x}^2}-3x\frac{dy}{dx}+y=\frac{sin(logx)+1}{x}logx.$ [2026/15m/6a]

2. Solve $\frac{d^{2}y}{d{x}^2}+a^{2}y=secax$ by the method of variation of parameters. [2026/15m/8a]

3. Find the complete solution of $x^3\frac{d^{3}y}{d{x}^3}+3x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=xlogx$. [2025/10m/7c(ii)]

4. Solve the differential equation $(x+2)\frac{d^{2}y}{d{x}^2}-(2x+5)\frac{dy}{dx}+2y=(1+x)e^x$ by the method of variation of parameters. [2025/15m/8a]

5. Find the second solution of the differential equation $xy″+(x-1)y\prime -y=0$ using $u\left(x\right)=-e^{-x}$ as one of the solutions. [2024/10m/6c(i)]

6. Find the general solution of the differential equation $x^{2}y″-2xy\prime +2y=x^{3}sinx$ by the method of variation of parameters. [2024/10m/6c(ii)]

7. Solve the following differential equation by using the method of variation of parameters : $(x^2-1)\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+2y={(x^2-1)}^2$, given that $y=x$ is one solution of the reduced equation. [2022/15m/6b]

8. Solve the following differential equation : ${(3x+2)}^2\frac{d^{2}y}{d{x}^2}+5(3x+2)\frac{dy}{dx}-3y=x^2+x+1$ [2022/10m/8a(ii)]

9. Solve the equation : $\frac{d^{2}y}{d{x}^2}+(tanx-3cosx)\frac{dy}{dx}+2y{cos}^{2}x={cos}^{4}x$ completely by demonstrating all the steps involved. [2021/15m/6b]

10. Find the general solution of the differential equation : $x^2\frac{d^{2}y}{d{x}^2}-2x(1+x)\frac{dy}{dx}+2(1+x)y=0.$ Hence, solve the differential equation : $x^2\frac{d^{2}y}{d{x}^2}-2x(1+x)\frac{dy}{dx}+2(1+x)y=x^3$ by the method of variation of parameters. [2021/10m/8a(ii)]

11. Using the method of variation of parameters, solve the differential equation $y″+(1-cotx)y\prime -ycotx={sin}^{2}x$, if $y=e^{-x}$ is one solution of CF. [2020/20m/6a]

12. Solve the following differential equation : ${(x+1)}^{2}y″-4(x+1)y\prime +6y=6{(x+1)}^2+sinlog(x+1)$ [2020/10m/8a(i)]

13. Solve the differential equation $\frac{d^{2}y}{d{x}^2}+(3sinx-cotx)\frac{dy}{dx}+2y{sin}^{2}x=e^{-cosx}{sin}^{2}x$ [2019/10m/6c(i)]

14. Find the linearly independent solutions of the corresponding homogeneous differential equation of the equation $x^{2}y″-2xy\prime +2y=x^{3}sinx$ and then find the general solution of the given equation by the method of variation of parameters. [2019/15m/7a]

15. Solve : ${(1+x)}^{2}y″+(1+x)y\prime +y=4cos(log(1+x))$ [2018/13m/7a]

16. Solve the differential equation : $x\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}-4x^{3}y=8x^{3}sin\left(x^2\right).$ [2017/9m/7b(i)]

17. Solve the following differential equation using method of variation of parameters : $\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}-2y=44-76x-48x^2.$ [2017/8m/7b(ii)]

18. Using the method of variation of parameters, solve the differential equation $(D^2+2D+1)y=e^{-x}log\left(x\right),[D\equiv \frac{d}{dx}]$ [2016/15m/6b]

19. Find the general solution of the equation $x^2\frac{d^{3}y}{d{x}^3}-4x\frac{d^{2}y}{d{x}^2}+6\frac{dy}{dx}=4$. [2016/15m/6c]

20. Solve : $x^4\frac{d^{4}y}{d{x}^4}+6x^3\frac{d^{3}y}{d{x}^3}+4x^2\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}-4y=x^2+2cos\left({log}_{e}x\right).$ [2015/13m/8d]

21. Solve by the method of variation of parameters : $\frac{dy}{dx}-5y=sinx$ [2014/10m/6a]

22. Solve the differential equation : $x^3\frac{d^{3}y}{d{x}^3}+3x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+8y=65cos\left({log}_{e}x\right)$ [2014/20m/6b]

23. Solve the following differential equation : $x\frac{d^{2}y}{d{x}^2}-2(x+1)\frac{dy}{dx}+(x+2)y=(x-2)e^{2x},$ when $e^x$ is a solution to its corresponding homogeneous differential equation. [2014/15m/7a]

24. Using the method of variation of parameters, solve the differential equation $\frac{d^{2}y}{d{x}^2}+a^{2}y=secax.$ [2013/10m/6b]

25. Find the general solution of the equation $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=lnxsin(lnx).$ [2013/15m/6c]

26. Solve the ordinary differential equation $x(x-1)y″-(2x-1)y\prime +2y=x^2(2x-3)$ [2012/20m/6c]

27. Using the method of variation of parameters, solve the second order differential equation $\frac{d^{2}y}{d{x}^2}+4y=tan2x.$ [2011/15m/6c]

28. Find the Wronskian of the set of functions $\{3x^3,\left|3x^3\right|\}$ on the interval $[-1,1]$ and determine whether the set is linearly dependent on $[-1,1]$. [2009/12m/5b]

29. Use the method of variation of parameters to find the general solution of $x^{2}y″-4xy\prime +6y=-x^{4}sinx.$ [2008/12m/5b]

30. Solve the differential equation $x^{3}y″-3x^{2}y\prime +xy=sin(lnx)+1.$ [2008/15m/6c]

4.5] Laplace transforms

21 questions

1. Apply Laplace transform to solve $(D^2+D)y\left(t\right)=2$, $D\equiv \frac{d}{dt}$, subject to the conditions $y\left(0\right)=3$, $y\prime \left(0\right)=1$. [2026/10m/5b]

2. If $L\{f\left(t\right)\}=\bar{f}\left(s\right)$, then show that $L\left\{\frac{f\left(t\right)}{t}\right\}={\int }_s^{\infty }\bar{f}\left(u\right)du$. Hence evaluate $L\left\{\frac{cosat-cosbt}{t}\right\}$. [2026/10m/7c(ii)]

3. If $F\left(s\right)$ and $G\left(s\right)$ are Laplace transforms of $f\left(t\right)$ and $g\left(t\right)$ respectively, then prove that $ℒ\left({\int }_0^{t}f\left(x\right)g(t-x)dx\right)=F\left(s\right)G\left(s\right)$. Using this result, solve the equation $y\left(t\right)=t+{\int }_0^{t}y\left(x\right)sin(t-x)dx$. [2025/15m/6a]

4. Solve the integral equation $y\left(t\right)=cost+{\int }_0^{t}y\left(x\right)cos(t-x)dx$ using Laplace transform. [2024/10m/5b]

5. Using Laplace transform, solve the initial value problem $y″+2y\prime +5y=\delta (t-2),y\left(0\right)=0,y\prime \left(0\right)=0$ where $\delta (t-2)$ denotes the Dirac delta function. [2024/15m/8a]

6. Given that $L\{f\left(t\right);p\}=F\left(p\right)$. Show that ${\int }_0^{\infty }\frac{f\left(t\right)}{t}dt={\int }_0^{\infty }F\left(x\right)dx$. Hence evaluate the integral ${\int }_0^{\infty }\frac{e^{-t}-e^{-3t}}{t}dt$. [2023/10m/5b]

7. Solve the following initial value problem by using Laplace transform technique : $\frac{d^{2}y}{d{t}^2}-4\frac{dy}{dt}+3y\left(t\right)=f\left(t\right),$ $y\left(0\right)=1$, $y\prime \left(0\right)=0$ and $f\left(t\right)$ is a given function of $t$. [2023/15m/8a]

8. Solve the following initial value problem by using Laplace’s transformation $\frac{d^{2}y}{d{t}^2}-3\frac{dy}{dt}+2y=h\left(t\right)$, where $h\left(t\right)=\{\begin{array}{cc}\hfill 2,\hfill & \hfill 0<t<4,\hfill \\ \hfill 0,\hfill & \hfill t>4,\hfill \end{array}y\left(0\right)=0,y\prime \left(0\right)=0$ [2022/15m/7b]

9. Solve the initial value problem : $\frac{d^{2}y}{d{x}^2}+4y=e^{-2x}sin2x;y\left(0\right)=y\prime \left(0\right)=0$ using Laplace transform method. [2021/10m/5b]

10. Using Laplace transform, solve the initial value problem $ty″+2ty\prime +2y=2$; $y\left(0\right)=1$ and $y\prime \left(0\right)$ is arbitrary. Does this problem have a unique solution? [2020/10m/7b]

11. Find the Laplace transforms of $t^{-1/2}$ and $t^{1/2}$. Prove that the Laplace transform of $t^{n+\frac{1}{2}}$, where $n\in \mathbb{N}$, is $\frac{\Gamma (n+1+\frac{1}{2})}{s^{n+1+\frac{1}{2}}}$ [2019/10m/6c(ii)]

12. (i) Find the Laplace transform of $f\left(t\right)=\frac{1}{\sqrt{t}}$. (ii) Find the inverse Laplace transform of $\frac{5s^2+3s-16}{(s-1)(s-2)(s+3)}$. [2018/10m/5d]

13. Solve the following initial value problem using Laplace transform : $\frac{d^{2}y}{d{x}^2}+9y=r\left(x\right),y\left(0\right)=0,y\prime \left(0\right)=4$ where $r\left(x\right)=\{\begin{array}{cc}\hfill 8sinx\hfill & \hfill \mathrm{\text{if}}0<x<\pi \hfill \\ \hfill 0\hfill & \hfill \mathrm{\text{if}}x\ge \pi \hfill \end{array}$ [2017/17m/8b]

14. Using Laplace transformation, solve the following : $y″-2y\prime -8y=0,y\left(0\right)=3,y\prime \left(0\right)=6$ [2016/10m/6d]

15. (i) Obtain Laplace Inverse transform of $\{ln(1+\frac{1}{s^2})+\frac{s}{s^2+25}{e}^{-\pi s}\}.$ \(ii\) Using Laplace transform, solve $y″+y=t$, $y\left(0\right)=1$, $y\prime \left(0\right)=-2$. [2015/12m/7a]

16. Solve the initial value problem $\frac{d^{2}y}{d{t}^2}+y=8e^{-2t}sint,y\left(0\right)=0,y\prime \left(0\right)=0$ by using Laplace-transform. [2014/20m/8c]

17. By using Laplace transform method, solve the differential equation $(D^2+n^2)x=asin(nt+\alpha )$, $D^2\equiv \frac{d^2}{d{t}^2}$ subject to the initial conditions $x=0$ and $\frac{dx}{dt}=0$, at $t=0$, in which $a$, $n$ and $\alpha$ are constants. [2013/15m/6d]

18. Using Laplace transforms, solve the initial value problem $y″+2y\prime +y=e^{-t},y\left(0\right)=-1,y\prime \left(0\right)=1$ [2012/12m/5c]

19. Use Laplace transform method to solve the following initial value problem : $\frac{d^{2}x}{d{t}^2}-2\frac{dx}{dt}+x=e^t,x\left(0\right)=2\mathrm{\text{and}}{\frac{dx}{dt}|}_{t=0}=-1.$ [2011/15m/6d]

20. Find the inverse Laplace transform of $F\left(s\right)=ln\left(\frac{s+1}{s+5}\right)$ [2009/20m/6b]

21. Using Laplace transform, solve the initial value problem $y″-3y\prime +2y=4t+e^{3t}$ with $y\left(0\right)=1$, $y\prime \left(0\right)=-1$. [2008/15m/6b]

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Last updated: 30 October 2026