IFoS PYQs 5
2004
1) Find the general solution of the partial differential equation \(\left(z^{2}-2 y z-y^{2}\right) p+x(y+z) q=x(y-z)\)
2) Apply charpit’s method to find the complete integral of the partial differential equation \(p x y+p q+q y=y z\).
3) Solve the Laplace equation \(\dfrac{\partial^{2} u}{\partial x^{2}}+\dfrac{\partial^{2} u}{\partial y^{2}}=0,0 \leq x \leq \pi, 0 \leq y \leq \infty\) satisfying the boundary conditions \(u(0, y)=0,\) for \(0 \leq y<\infty\) \(u(\pi, y)=0,\) for \(0<y<\infty\) \(u(x, \infty)=0,\) for \(0<x<\pi \quad\) and \(u(x, 0)=u_{0},\) for \(0<x<\pi\)
2003
1) Find the general solution of the partial differential equation \((m z-n y) \dfrac{\partial z}{\partial x}+(n x-l z) \dfrac{\partial z}{\partial y}=l y-m x\).
2) Form the partial differential equation by eliminating the arbitrary function from \(f\left(x^{2}+y^{2}, z-x y\right)=0, z=z(z, y)\).
3) Solve \(\dfrac{\partial u}{\partial t}=c^{2} \dfrac{\partial^{2} u}{\partial t^{2}}\) given that (i) \(u =0,\) When \(t=0 t=0\) for all \(t\) (ii) \(u =0,\) When \(x=l\) for all \(t\) (iii) \(\left.\begin{array}{rl} u & =x \quad \text { in }(0,l/ 2) \\ & =l-x \text { in }(l / 2, l) \end{array}\right\} \text { at } t=0\)
2002
1) Solve completely \(\dfrac{x}{p q}=\dfrac{x}{q}+\dfrac{y}{p}+\sqrt{p q}\).
2) Using charpit’s method solve completely \(p^{2}-q^{2}=(x+y)^{2}\).
3) Obtain the general solution of the following equation \(\dfrac{\partial^{2} z}{\partial x^{2}}-3 \dfrac{\partial^{2} z}{\partial x \partial y}+2 \dfrac{\partial^{2} z}{\partial y^{2}}=(2+4 x) e^{x-2 y}\).
2001
1) Find the complete integral of the partial differential equation \(x^{2} p^{2}+y^{2} q^{2}=z^{2}\).
2) Solve by Charpit’s method \(\left(p^{2}+q^{2}\right) y=q z\).
3) If \(\varphi( x )\) is a continuous and bounded function for \(-\infty<x<\infty,\) prove that the function \(u(x, t)=\dfrac{1}{2 \sqrt{\pi k t}} \int_{-\infty}^{\infty} \varphi(\xi) e^{-(x-\xi)^{2} / 4 x} d \xi\) is a solution of the initial value problem: \(u_{t}-k u_{x x}=0,-\infty<x<\infty, t>0\) \(u(x, 0)=\varphi(x)\) for \(-\infty<x<\infty\).
2000
1) Solve the following initial value problem \((y+z) z_{x}+y z_{y}=x-y ; z=1+t\) on die initial curve \(C : x=t, y=1 ;-\infty<T<\infty\).
2) Determine the complete integral of the equation \(z \dfrac{\partial z}{\partial x} \dfrac{\partial z}{\partial y}=\left(\dfrac{\partial z}{\partial x}\right)^{2}\left(x \dfrac{\partial z}{\partial y}+\left[\dfrac{\partial z}{\partial x}\right]^{2}\right)+\left(\dfrac{\partial z}{\partial y}\right)^{2}\left(y \dfrac{\partial z}{\partial x}+\left[\dfrac{\partial z}{\partial y}\right]^{2}\right)\).
3) Solve the following partial differential equation \(\dfrac{\partial^{2} z}{\partial x^{2}}+\dfrac{\partial^{2} z}{\partial x \partial y}-2 \dfrac{\partial^{2} z}{\partial y^{2}}-\dfrac{\partial z}{\partial x}-2 \dfrac{\partial z}{\partial y}=0\).