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

Here are all 86 Partial Differential Equations questions asked in UPSC Mathematics Optional Paper II from 2008 to 2026, arranged under the 4 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.

5.1] Formulation of PDEs

9 questions

1. By eliminating the arbitrary functions $f$ and $g$ from $z=f(x^2-y)+g(x^2+y)$, form partial differential equation. [2023/10m/5a]

2. It is given that the equation of any cone with vertex at $(a,b,c)$ is $f(\frac{x-a}{z-c},\frac{y-b}{z-c})=0$. Find the differential equation of the cone. [2022/10m/5a]

3. Obtain the partial differential equation by eliminating arbitrary function $f$ from the equation $f(x+y+z,x^2+y^2+z^2)=0$. [2021/10m/5a]

4. Form a partial differential equation by eliminating the arbitrary functions $f\left(x\right)$ and $g\left(y\right)$ from $z=yf\left(x\right)+xg\left(y\right)$ and specify its nature (elliptic, hyperbolic or parabolic) in the region $x>0$, $y>0$. [2020/10m/5a]

5. Form a partial differential equation of the family of surfaces given by the following expression : $\psi (x^2+y^2+2z^2,y^2-2zx)=0$. [2019/10m/5a]

6. Find the partial differential equation of the family of all tangent planes to the ellipsoid : $x^2+4y^2+4z^2=4$, which are not perpendicular to the $xy$ plane. [2018/10m/5a]

7. Form a partial differential equation by eliminating the arbitrary functions $f$ and $g$ from $z=yf\left(x\right)+xg\left(y\right)$. [2013/10m/5a]

8. Show that the differential equation of all cones which have their vertex at the origin is $px+qy=z$. Verify that this equation is satisfied by the surface $yz+zx+xy=0$. [2009/12m/5a]

9. Form the partial differential equation by eliminating the arbitrary function $f$ given by : $f(x^2+y^2,z-xy)=0.$ [2009/6m/5b(i)]

5.2] First order PDEs

22 questions

1. Find the integral surface of the equations $\frac{dx}{xz-y}=\frac{dy}{yz-x}=\frac{dz}{1-z^2}$. [2026/10m/5a]

2. Find the complete integral of $z(p^2-q^2)=x-y$; $p\equiv \frac{\partial z}{\partial x}$, $q\equiv \frac{\partial z}{\partial y}$. [2025/15m/7a]

3. Find the characteristics of the partial differential equation $p^2+q^2=2;p\equiv \frac{\partial z}{\partial x},q\equiv \frac{\partial z}{\partial y}$ and determine the integral surface which passes through $x=0$, $z=y$. [2025/15m/8a]

4. Find the integral surface of the following quasi-linear equation $(y-\varphi )\frac{\partial \varphi }{\partial x}+(\varphi -x)\frac{\partial \varphi }{\partial y}=x-y,$ which passes through the curve $\varphi =0$, $xy=1$ and through the circle $x+y+\varphi =0$, $x^2+y^2+{\varphi }^2=a^2$. [2024/15m/7a]

5. Find a complete integral of the partial differential equation $p={(z+qy)}^2$ by using Charpit’s method. [2021/15m/8a]

6. Find the integral surface of the partial differential equation : $(x-y)y^2\frac{\partial z}{\partial x}+(y-x)x^2\frac{\partial z}{\partial y}=(x^2+y^2)z$ that contains the curve : $xz=a^3$, $y=0$ on it. [2020/15m/6a]

7. Find the solution of the partial differential equation : $z=\frac{1}{2}(p^2+q^2)+(p-x)(q-y);p\equiv \frac{\partial z}{\partial x},q\equiv \frac{\partial z}{\partial y}$ which passes through the $x$-axis. [2020/15m/7a]

8. Solve the first order quasilinear partial differential equation by the method of characteristics : $x\frac{\partial u}{\partial x}+(u-x-y)\frac{\partial u}{\partial y}=x+2y$ in $x>0,-\infty <y<\infty$ with $u=1+y$ on $x=1$. [2019/15m/6a]

9. Find the general solution of the partial differential equation : $(y^{3}x-2x^4)p+(2y^4-x^{3}y)q=9z(x^3-y^3)$, where $p=\frac{\partial z}{\partial x}$, $q=\frac{\partial z}{\partial y}$, and find its integral surface that passes through the curve : $x=t$, $y=t^2$, $z=1$. [2018/15m/6a]

10. Find a complete integral of the partial differential equation $2(pq+yp+qx)+x^2+y^2=0.$ [2017/15m/6a]

11. Find the general equation of surfaces orthogonal to the family of spheres given by $x^2+y^2+z^2=cz$. [2016/10m/5a]

12. Find the general integral of the partial differential equation $(y+zx)p-(x+yz)q=x^2-y^2.$ [2016/10m/5e]

13. Determine the characteristics of the equation $z=p^2-q^2$, and find the integral surface which passes through the parabola $4z+x^2=0,y=0$. [2016/15m/6a]

14. Solve the partial differential equation $(y^2+z^2-x^2)p-2xyq+2xz=0$ where $p=\frac{\partial z}{\partial x}$ and $q=\frac{\partial z}{\partial y}$. [2015/10m/5a]

15. Solve for the general solution $pcos(x+y)+qsin(x+y)=z$, where $p=\frac{\partial z}{\partial x}$ and $q=\frac{\partial z}{\partial y}$. [2015/15m/6a]

16. Find the surface which intersects the surfaces of the system $z(x+y)=C(3z+1),\left(C\mathrm{\text{being a constant}}\right)$ orthogonally and which passes through the circle $x^2+y^2=1$, $z=1$. [2013/15m/6b]

17. Solve the partial differential equation $px+qy=3z$. [2012/20m/6a]

18. Solve the PDE $(x+2z)\frac{\partial z}{\partial x}+(4zx-y)\frac{\partial z}{\partial y}=2x^2+y$ [2011/12m/5b]

19. Solve the following partial differential equation $zp+yq=x$ $x_0\left(s\right)=s,y_0\left(s\right)=1,z_0\left(s\right)=2s$ by the method of characteristics. [2010/20m/6a]

20. Find the integral surface of : $x^{2}p+y^{2}q+z^2=0$ which passes through the curve : $xy=x+y$, $z=1$. [2009/6m/5b(ii)]

21. Find the general solution of the partial differentia. equation $(2xy-1)p+(z-2x^2)q=2(x-yz)$ and also find the particular solution which passes through the lines $x=1$, $y=0$. [2008/12m/5a]

22. Find complete and singular integrals of $2xz-p{x}^2-2qxy+pq=0$ using Charpit’s method. [2008/15m/6b]

5.3] Second order linear PDEs

33 questions

1. Reduce the partial differential equation $y\frac{{\partial }^{2}z}{\partial x^2}+(x+y)\frac{{\partial }^{2}z}{\partial x\partial y}+x\frac{{\partial }^{2}z}{\partial y^2}=0$ to canonical form and hence solve it. [2026/20m/6a]

2. Solve the partial differential equation $(2D^2-5DD\prime +2D{\prime }^2)z=24(y-x)+sin(y-x)$ where $D\equiv \frac{\partial }{\partial x}$, $D\prime \equiv \frac{\partial }{\partial y}$, $z=z(x,y)$. [2026/15m/7a]

3. Find the solution of the equation $(D^2+DD\prime -2D^2)z=ysinx$, where $D\equiv \frac{\partial }{\partial x}$ and $D\prime \equiv \frac{\partial }{\partial y}$. [2025/10m/5a]

4. Show that if $f$ and $g$ are arbitrary functions of their respective arguments, then $u=f(x-kt+i\alpha y)+g(x-kt-i\alpha y)$, is a solution of $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=\frac{1}{C^2}\frac{{\partial }^{2}u}{\partial t^2},$ where ${\alpha }^2=1-\frac{k^2}{C^2}$. [2024/10m/5a]

5. Solve the partial differential equation $\frac{\partial }{\partial y}(\frac{\partial \varphi }{\partial x}+\varphi )+2x^{2}y(\frac{\partial \varphi }{\partial x}+\varphi )=0$ by transforming it to the canonical form. [2024/15m/8a]

6. Find the surface passing through the two lines $z=x=0$ and $z-1=x-y=0$, and satisfying the partial differential equation $\frac{{\partial }^{2}z}{\partial x^2}-4\frac{{\partial }^{2}z}{\partial x\partial y}+4\frac{{\partial }^{2}z}{\partial y^2}=0$. [2023/15m/6a]

7. Reduce the partial differential equation $\frac{{\partial }^{2}z}{\partial y^2}-\frac{{\partial }^{2}z}{\partial x\partial y}+\frac{\partial z}{\partial x}-\frac{\partial z}{\partial y}(1+\frac{1}{x})+\frac{z}{x}=0$ to canonical form. [2023/15m/8a]

8. Find the general solution of the partial differential equation $(D^2+DD\prime -6D{\prime }^2)z=x^{2}sin(x+y)$ where $D\equiv \frac{\partial }{\partial x}$ and $D\prime \equiv \frac{\partial }{\partial y}$. [2022/15m/7a]

9. Reduce the following partial differential equation to a canonical form and hence solve it : $y{u}_{xx}+(x+y)u_{xy}+x{u}_{yy}=0$ [2022/15m/8a]

10. Find the general solution of the partial differential equation $(D^2-D{\prime }^2-3D+3D\prime )z=xy+e^{x+2y}$ where $D\equiv \frac{\partial }{\partial x}$ and $D\prime \equiv \frac{\partial }{\partial y}$. [2021/15m/7a]

11. Form a partial differential equation by eliminating the arbitrary functions $f\left(x\right)$ and $g\left(y\right)$ from $z=yf\left(x\right)+xg\left(y\right)$ and specify its nature (elliptic, hyperbolic or parabolic) in the region $x>0$, $y>0$. [2020/10m/5a]

12. Solve the partial differential equation : $(D^3-2D^{2}D\prime -DD{\prime }^2+2D{\prime }^3)z=e^{2x+y}+sin(x-2y);$ $D\equiv \frac{\partial }{\partial x},D\prime \equiv \frac{\partial }{\partial y}$ [2020/10m/5d]

13. Reduce the following second order partial differential equation to canonical form and find the general solution : $\frac{{\partial }^{2}u}{\partial x^2}-2x\frac{{\partial }^{2}u}{\partial x\partial y}+x^2\frac{{\partial }^{2}u}{\partial y^2}=\frac{\partial u}{\partial y}+12x.$ [2019/20m/7c]

14. Solve the partial differential equation : $(2D^2-5DD\prime +2D{\prime }^2)z=5sin(2x+y)+24(y-x)+e^{3x+4y}$ where $D\equiv \frac{\partial }{\partial x}$, $D\prime \equiv \frac{\partial }{\partial y}$. [2018/15m/7a]

15. Solve $(D^2-2DD\prime +D{\prime }^2)z=e^{x+2y}+x^3+sin2x$, where $D\equiv \frac{\partial }{\partial x},D\prime \equiv \frac{\partial }{\partial y},D^2\equiv \frac{{\partial }^2}{\partial x^2},D{\prime }^2\equiv \frac{{\partial }^2}{\partial y^2}.$ [2017/10m/5a]

16. Reduce the equation $y^2\frac{{\partial }^{2}z}{\partial x^2}-2xy\frac{{\partial }^{2}z}{\partial x\partial y}+x^2\frac{{\partial }^{2}z}{\partial y^2}=\frac{y^2}{x}\frac{\partial z}{\partial x}+\frac{x^2}{y}\frac{\partial z}{\partial y}$ to canonical form and hence solve it. [2017/15m/7a]

17. Solve the partial differential equation $\frac{{\partial }^{3}z}{\partial x^3}-2\frac{{\partial }^{3}z}{\partial x^2\partial y}-\frac{{\partial }^{3}z}{\partial x\partial y^2}+2\frac{{\partial }^{3}z}{\partial y^3}=e^{x+y}$ [2016/15m/7a]

18. Solve $(D^2+DD\prime -2D{\prime }^2)u=e^{x+y}$, where $D=\frac{\partial }{\partial x}$ and $D\prime =\frac{\partial }{\partial y}$. [2015/10m/5b]

19. Reduce the second-order partial differential equation $x^2\frac{{\partial }^{2}u}{\partial x^2}-2xy\frac{{\partial }^{2}u}{\partial x\partial y}+y^2\frac{{\partial }^{2}u}{\partial y^2}+x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=0$ into canonical form. Hence, find its general solution. [2015/15m/8a]

20. Solve the partial differential equation $(2D^2-5DD\prime +2D{\prime }^2)z=24(y-x)$. [2014/10m/5a]

21. Reduce the equation $\frac{{\partial }^{2}z}{\partial x^2}=x^2\frac{{\partial }^{2}z}{\partial y^2}$ to canonical form. [2014/15m/6a]

22. Reduce the equation $y\frac{{\partial }^{2}z}{\partial x^2}+(x+y)\frac{{\partial }^{2}z}{\partial x\partial y}+x\frac{{\partial }^{2}z}{\partial y^2}=0$ to its canonical form when $x\ne y$. [2013/10m/5b]

23. Solve $(D^2+DD\prime -6D{\prime }^2)z=x^{2}sin(x+y)$ where $D$ and $D\prime$ denote $\frac{\partial }{\partial x}$ and $\frac{\partial }{\partial y}$. [2013/15m/6a]

24. Solve the partial differential equation $(D-2D\prime ){(D-D\prime )}^{2}z=e^{x+y}.$ [2012/12m/5a]

25. Solve the PDE $(D^2-D{\prime }^2+D+3D\prime -2)z=e^{(x-y)}-x^{2}y$ [2011/12m/5a]

26. Find the surface satisfying $\frac{{\partial }^{2}z}{\partial x^2}=6x+2$ and touching $z=x^3+y^3$ along its section by the plane $x+y+1=0$. [2011/20m/6a]

27. Solve the $PDE$ $(D^2-D\prime )(D-2D\prime )Z=e^{2x+y}+xy.$ [2010/12m/5a]

28. Find the surface satisfying the $PDE$ $(D^2-2DD\prime +D{\prime }^2)Z=0$ and the conditions that $bZ=y^2$ when $x=0$ and $aZ=x^2$ when $y=0$. [2010/12m/5b]

29. Reduce the following 2nd order partial differential equation into canonical form and find its general solution $x{u}_{xx}+2x^2{u}_{xy}-u_x=0.$ [2010/20m/6b]

30. Find the characteristics of : $y^{2}r-x^{2}t=0$ where $r$ and $t$ have their usual meanings. [2009/15m/6a]

31. Solve : $(D^2-DD\prime -2D{\prime }^2)z=(2x^2+xy-y^2)sinxy-cosxy$ where $D$ and $D\prime$ represent $\frac{\partial }{\partial x}$ and $\frac{\partial }{\partial y}$. [2009/15m/6b]

32. Find general solution of the particle differential equation : $(D^2+DD\prime -6D{\prime }^2)z=ycosx,$ where $D=\frac{\partial }{\partial x}$, $D\prime =\frac{\partial }{\partial y}$. [2008/12m/5b]

33. Reduce $\frac{{\partial }^{2}z}{\partial x^2}=x^2\frac{{\partial }^{2}z}{\partial y^2}$ to canonical form. [2008/15m/6c]

5.4] Wave, heat and Laplace equations

22 questions

1. Two-dimensional harmonic equation in plane polar coordinates $(r,\theta )$ takes the form : $\frac{{\partial }^{2}V}{\partial r^2}+\frac{1}{r}\frac{\partial V}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}V}{\partial {\theta }^2}=0$ Deduce that it has solutions of the form $(A{r}^n+B{r}^{-n})e^{\pm in\theta }$, where $A$, $B$, $n$ are constants. Determine $V$ if it satisfies the two-dimensional harmonic equation in the region $0\le r\le a$, $0\le \theta \le 2\pi$ and satisfies the conditions : (i) $V$ remains finite as $r\to 0$, (ii) $V={\sum }_n^{}{c}_{n}cos\left(n\theta \right)$ on $r=a$. [2026/15m/8a]

2. Solve $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$ for a rectangular plate subject to the boundary conditions $\begin{array}{cc}\hfill \hfill & \hfill u(0,y)=0,u(a,y)=0\hfill \\ \hfill \hfill & \hfill u(x,0)=0,u(x,b)=f\left(x\right)\hfill \end{array}$ [2025/20m/6a]

3. Show that the solution of the two-dimensional Laplace’s equation $\frac{{\partial }^2\varphi (x,y)}{\partial x^2}+\frac{{\partial }^2\varphi (x,y)}{\partial y^2}=0,x\in (-\infty ,\infty ),y\ge 0$ subject to the boundary condition $\varphi (x,0)=f\left(x\right)$, $x\in (-\infty ,\infty )$, along with $\varphi (x,y)\to 0$ for $\left|x\right|\to \infty$ and $y\to \infty$ can be written in the form $\varphi (x,y)=\frac{y}{\pi }{\int }_{-\infty }^{\infty }\frac{f\left(\xi \right)d\xi }{y^2+{(x-\xi )}^2}.$ [2024/20m/6a]

4. Solve the partial differential equation $a^2\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}u}{\partial t^2},0<x<L,t>0$ subject to the conditions $\begin{array}{cc}\hfill \hfill & \hfill u(0,t)=0,u(L,t)=0,t>0\hfill \\ \hfill \hfill & \hfill u(x,0)=x,{\left(\frac{\partial u}{\partial t}\right)}_{t=0}=1,0<x<L\hfill \end{array}$ [2023/20m/7c]

5. Solve the heat equation $\frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial x^2}$, $0<x<l$, $t>0$ subject to the conditions $\begin{array}{cc}\hfill \hfill & \hfill u(0,t)=u(l,t)=0\hfill \\ \hfill \hfill & \hfill u(x,0)=x(l-x),0\le x\le l\hfill \end{array}$ [2022/20m/6a]

6. Solve the wave equation $a^2\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}u}{\partial t^2},0<x<L,t>0$ subject to the conditions $\begin{array}{cc}\hfill \hfill & \hfill u(0,t)=0,u(L,t)=0\hfill \\ \hfill \hfill & \hfill u(x,0)=\frac{1}{4}x(L-x),{\frac{\partial u}{\partial t}|}_{t=0}=0\hfill \end{array}$ [2021/20m/6a]

7. One end of a tightly stretched flexible thin string of length $l$ is fixed at the origin and the other at $x=l$. It is plucked at $x=\frac{l}{3}$ so that it assumes initially the shape of a triangle of height $h$ in the $x$–$y$ plane. Find the displacement $y$ at any distance $x$ and at any time $t$ after the string is released from rest. Take, $\frac{\mathrm{\text{horizontal tension}}}{\mathrm{\text{mass per unit length}}}=c^2$. [2020/20m/8a]

8. A thin annulus occupies the region $0<a\le r\le b$, $0\le \theta \le 2\pi$. The faces are insulated. Along the inner edge the temperature is maintained at $0^{\circ }$, while along the outer edge the temperature is held at $T=Kcos\frac{\theta }{2}$, where $K$ is a constant. Determine the temperature distribution in the annulus. [2018/20m/8c]

9. Let $\Gamma$ be a closed curve in $xy$-plane and let $S$ denote the region bounded by the curve $\Gamma$. Let $\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}=f(x,y)\forall (x,y)\in S.$ If $f$ is prescribed at each point $(x,y)$ of $S$ and $w$ is prescribed on the boundary $\Gamma$ of $S$, then prove that any solution $w=w(x,y)$, satisfying these conditions, is unique. [2017/10m/5d]

10. Given the one-dimensional wave equation $\frac{{\partial }^{2}y}{\partial t^2}=c^2\frac{{\partial }^{2}y}{\partial x^2};t>0,$ where $c^2=\frac{T}{m}$, $T$ is the constant tension in the string and $m$ is the mass per unit length of the string. (i) Find the appropriate solution of the above wave equation. (ii) Find also the solution under the conditions $y(0,t)=0$, $y(l,t)=0$ for all $t$ and ${\left[\frac{\partial y}{\partial t}\right]}_{t=0}=0$, $y(x,0)=asin\frac{\pi x}{l}$, $0<x<l$, $a>0$. [2017/20m/8a]

11. Find the temperature $u(x,t)$ in a bar of silver of length 10 cm and constant cross-section of area 1 cm${}^2$. Let density $\rho =10.6$ g/cm${}^3$, thermal conductivity $K=1.04$ cal/(cm sec ${}^{\circ }$C) and specific heat $\sigma =0.056$ cal/g ${}^{\circ }$C. The bar is perfectly isolated laterally, with ends kept at $0^{\circ }$C and initial temperature $f\left(x\right)=sin\left(0.1\pi x\right){}^{\circ }$C. Note that $u(x,t)$ follows the heat equation $u_t=c^2{u}_{xx}$, where $c^2=K/\left(\rho \sigma \right)$. [2016/20m/8a]

12. Find the solution of the initial-boundary value problem $\begin{array}{cccc}\hfill \hfill & \hfill u_t-u_{xx}+u=0,\hfill & \hfill \hfill & \hfill 0<x<l,t>0\hfill \\ \hfill \hfill & \hfill u(0,t)=u(l,t)=0,\hfill & \hfill \hfill & \hfill t\ge 0\hfill \\ \hfill \hfill & \hfill u(x,0)=x(l-x),\hfill & \hfill \hfill & \hfill 0<x<l\hfill \end{array}$ [2015/15m/7a]

13. Find the deflection of a vibrating string (length $=\pi$, ends fixed, $\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}$) corresponding to zero initial velocity and initial deflection $f\left(x\right)=k(sinx-sin2x)$ [2014/15m/7a]

14. Solve $\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}$, $0<x<1$, $t>0$, given that (i) $u(x,0)=0$, $0\le x\le 1$ (ii) $\frac{\partial u}{\partial t}(x,0)=x^2$, $0\le x\le 1$ (iii) $u(0,t)=u(1,t)=0$, for all $t$ [2014/15m/8a]

15. A tightly stretched string with fixed end points $x=0$ and $x=l$ is initially at rest in equilibrium position. If it is set vibrating by giving each point a velocity $\lambda \cdot x(l-x)$, find the displacement of the string at any distance $x$ from one end at any time $t$. [2013/20m/6c]

16. A string of length $l$ is fixed at its ends. The string from the mid-point is pulled up to a height $k$ and then released from rest. Find the deflection $y(x,t)$ of the vibrating string. [2012/20m/6b]

17. The edge $r=a$ of a circular plate is kept at temperature $f\left(\theta \right)$. The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in steady state. [2012/20m/7b]

18. Solve $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0,0\le x\le a,0\le y\le b$ satisfying the boundary conditions $u(0,y)=0,u(x,0)=0,u(x,b)=0$ $\frac{\partial u}{\partial x}(a,y)=T{sin}^3\frac{\pi y}{a}.$ [2011/20m/6b]

19. Obtain temperature distribution $y(x,t)$ in a uniform bar of unit length whose one end is kept at ${10}^{\circ }$C and the other end is insulated. Also it is given that $y(x,0)=1-x$, $0<x<1$. [2011/20m/6c]

20. Solve the following heat equation $\begin{array}{cc}\hfill \hfill & \hfill u_t-u_{xx}=0,0<x<2,t>0\hfill \\ \hfill \hfill & \hfill u(0,t)=u(2,t)=0,t>0\hfill \\ \hfill \hfill & \hfill u(x,0)=x(2-x),0\le x\le 2.\hfill \end{array}$ [2010/20m/6c]

21. A tightly stretched string has its ends fixed at $x=0$ and $x=l$. At time $t=0$, the string is given a shape defined by $f\left(x\right)=\mu x(l-x)$, where $\mu$ is a constant, and then released. Find the displacement of any point $x$ of the string at time $t>0$. [2009/30m/6c]

22. Find the steady state temperature distribution in a thin rectangular plate bounded by the lines $x=0$, $x=a$, $y=0$ and $y=b$. The edges $x=0$, $x=a$ and $y=0$ are kept at temperature zero while the edge $y=b$ is kept at ${100}^{\circ }$C. [2008/30m/6a]

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Last updated: 30 October 2026