PYQs

1) Solve the partial differential equation:

where $D\equiv {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}$, $D^{\prime }\equiv {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$.

[2018, 15M]

2) Solve $(D^2-2D{D}^{\mathrm{\prime }}-D^2)z$=$e^{x+2y}+x^3+\sin 2x$ where $D\equiv {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}$, $D^{\mathrm{\prime }}\equiv {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$, $D^2\equiv {\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}}$, $D^{\mathrm{\prime }2}\equiv {\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}}$.

[2017, 10M]

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

[2017, 10M]

4) Solve the partial differential equation ${\displaystyle \frac{{\mathrm{\partial }}^{3}z}{\mathrm{\partial }{x}^3}}-2{\displaystyle \frac{{\mathrm{\partial }}^{3}z}{\mathrm{\partial }{x}^2\mathrm{\partial }y}}{\displaystyle \frac{{\mathrm{\partial }}^{3}z}{\mathrm{\partial }x\mathrm{\partial }{y}^2}}+2{\displaystyle \frac{{\mathrm{\partial }}^{3}z}{\mathrm{\partial }{y}^3}}=e^{x+y}$.

[2016, 15M]

5) Solve: $(D^2+D{D}^{\mathrm{\prime }}-2D^{\mathrm{\prime }})u=e^{x+y}$, where $D={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}$ and $D^{\mathrm{\prime }}={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$.

[2015, 15M]

6) Solve the partial differential equation $(2D^2-5D{D}^{\mathrm{\prime }}+2D^{\mathrm{\prime }2})z=24(y-x)$.

[2014, 10M]

7) Solve $(D^2+D{D}^{\mathrm{\prime }}-6D^{\mathrm{\prime }2})z=x^2\sin (x+y)$ where $D$ and $D^{\mathrm{\prime }}$ denote ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}$ and ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$.

[2013, 15M]

8) Solve the PDE $(D^2-D^{\mathrm{\prime }2}+D+3D^{\mathrm{\prime }}-2)z$=$e^{(x-y)}-x^{2}y$.

[2011, 12M]

9) Solve the PDE $(D^2-D^{\mathrm{\prime }})(D-2D^{\mathrm{\prime }})Z=e^{2x+y}+xy$.

[2010, 12M]

10) Solve: $(D^2-D{D}^{\mathrm{\prime }}-2D^{\mathrm{\prime }2})z$=$(2x^2+xy-y^2)\sin xy-\cos xy$ where $D$ and $D^{\mathrm{\prime }}$ represent ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}$ and ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$ and ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$.

[2009, 15M]

11) Find the general solution of the partial differential equation: $(D^2+D{D}^{\mathrm{\prime }}-6D^2)z=y\cos x$, where $D\equiv {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}$, $D^{\mathrm{\prime }}\equiv {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$.

[2008, 12M]

12) Solve: ${\displaystyle \frac{{\mathrm{\partial }}^{3}z}{\mathrm{\partial }{x}^3}}-4{\displaystyle \frac{{\mathrm{\partial }}^{3}z}{\mathrm{\partial }{x}^2\mathrm{\partial }y}}+4{\displaystyle \frac{{\mathrm{\partial }}^{3}z}{\mathrm{\partial }x\mathrm{\partial }{y}^2}}$=$2\sin (3x+2y)$.

[2006, 12M]

13) Solve the partial differential equation: ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}-{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}-2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=(y-1)e^x$.

[2004, 15M]

14) Find the general solution of ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}+3{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=x+y+\cos (2x+3y)$.

[2003, 12M]

1) Reduce the following second order partial differential equation to canonical from and find the general solution:

${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}-2x{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}}+x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}$=${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}+12x$

[2019, 20M]

2) Reduce the equation $y^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}-2xy{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}$=${\displaystyle \frac{y^2}{\mathrm{\partial }x}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}}+{\displaystyle \frac{x^2}{\mathrm{\partial }y}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}$ to canonical form and hence solve it.

[2017, 15M]

3) Find the solution of the initial-boundary value problem:
$u_t-u_{xx}+u=0$, $0<x<l$, $t>0$ $u(0,t)=u(l,t)=0$, $t\ge 0$
$u(x,0)=x(l-x)$, $0<x<l$

[2015, 15M]

4) Reduce the second-order partial differential equation $x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}-2xy{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}}+y^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}+x{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}+y{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}=0$ into canonical form. Hence, find its general solution.

[2015, 15M]

5) Reduce the equation ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}=x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}$ to canonical form.

[2014, 15M]

6) Reduce the equation $y{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}+(x+y){\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+x{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=0$ to its canonical from when $x\ne y$.

[2013, 10M]

7) Reduce the following $2^{nd}$ order partial differential equation into canonical form and find find its general solution.
$x{u}_{xx}+2x^2{u}_{xy}-u_x=0$.

[2010, 20M]

8) Reduce ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}=x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}$ canonical form.

[2008, 15M]

9) Solve the equation $x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}-y^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}+x{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}}-y{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}=x^2{y}^4$ by reducing it to the equation with constant coefficients.

[2001, 20M]