2008

2007

1) Find the integral curves of the equations \(\dfrac{d x}{(x+z)}=\dfrac{d y}{y}=\dfrac{d x}{\left(z+y^{2}\right)}\)

[10M]

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

[10M]

3) Solve \(\dfrac{\partial^{2} z}{\partial x^{2}}-\dfrac{\partial^{2} z}{\partial y^{2}}=x-y\)

[10M]

2006

1) Apply Charpit’s method to solve the equation \(2 z+p^{2}+q y+2 y^{2}=0\)

[10M]

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

[10M]

3) Solve: \(\dfrac{\partial^{2} z}{\partial x^{2}}-3 \dfrac{\partial^{2} z}{\partial x \partial y}+2 \dfrac{\partial^{2} z}{\partial y^{2}}=e^{2 x+3 y}+\sin (x-2 y)\)

[10M]

2005

1) Apply Charpit’s method to solve the equation \(2 z+p^{2}+q y+2 y^{2}=0\)

[10M]

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

[10M]

3) Solve: \(\dfrac{\partial^{2} z}{\partial x^{2}}+\dfrac{\partial^{2} z}{\partial x \partial y}-6 \dfrac{\partial^{2} z}{\partial y^{2}}=y \cos x\)

[10M]