2008

2007

1) Find the integral curves of the equations ${\displaystyle \frac{dx}{(x+z)}}={\displaystyle \frac{dy}{y}}={\displaystyle \frac{dx}{(z+y^2)}}$

[10M]

2) Find the complete integral of $p^{2}x+q^{2}y=z$

[10M]

3) Solve ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}-{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=x-y$

[10M]

2006

1) Apply Charpit’s method to solve the equation $2z+p^2+qy+2y^2=0$

[10M]

2) Solve ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}}=c^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$ given that (i) $u=0$ when $x=0$ for all $t$ (ii) $u=0$ when $x=$ for all $t$

[10M]

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

[10M]

2005

1) Apply Charpit’s method to solve the equation $2z+p^2+qy+2y^2=0$

[10M]

2) Solve ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}}=c^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$ given that
(i) $u=0,$ when $x0$ for all $t$ (ii) $u=0,$ when $x=1$ for all $t$ (iii)\left \begin{array}{l} u=\dfrac{b x}{a}, 0< x < \alpha \\ 
=\dfrac{b(t-x)}{1-a}, \quad a< x < \end{array}\right\}$\text{left begin{array}{l} u=dfrac{b x}{a}, 0< x < alpha =dfrac{b(t-x)}{1-a}, quad a< x < end{array}right}}$ at $t=0$ (iv) ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}=0$ at $t=0,x$ in $(0,b)$

[10M]

3) Solve: ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}-6{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=y\cos x$

[10M]