2000

1) Solve: $pq=x^m{y}^n{z}^{2l}$

[12M]

2) Prove that if $x_1^3+x_2^3+x_3^3=1$ when $z=0,$ the solution of the equation $(\mathrm{S}-{\mathrm{x}}_1){\mathrm{p}}_1+(\mathrm{S}-{\mathrm{x}}_2){\mathrm{p}}_2+(\mathrm{S}-{\mathrm{x}}_3){\mathrm{p}}_3=\mathrm{S}-\mathrm{z}$ can be given in the form $S^3{\{{(x_1-z)}^3+{(x_2-z)}^3+{(x_3-z)}^3\}}^4={(x_1+x_2+x_3-3z)}^3$ where $S=x_1+x_2+x_3+z$ and $p_i={\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_i}},i=1,2,3$

[12M]

3) Solve by Charpit’s method the equation $p^{2}x(x-1)+2pqxy+q^{2}y(y-1)-2pxz-2qyz+z^2=0$

[15M]

4) Solve: $({\mathrm{D}}^2-\mathrm{D}{D}^{\mathrm{\prime }}-2{\mathrm{D}}^2)\mathrm{z}=2\mathrm{x}+3\mathrm{y}+{\mathrm{e}}^{3\mathrm{x}+4\mathrm{y}}$

[15M]

5) A tightly stretched string with fixed end points $x=0,x=l$ is initially at rest in equilibrium position. If it is set vibrating by giving each point $x$ of it a velocity $kx(l-x),$ obtain at time $t$ the displacement y at a distance $x$ from the end $x=0$.

[30M]

1999

1) Verify that the differential equation $(y^2+yz)dx+(xz+z^2)dy+(y^2-xy)dz=0$
is integrable and find its primitive.

[20M]

2) Find the surface which intersects the surface of the system $z(x+y)=c(3z+1),c=$a constant,orthogonally and which passes through the circle $x^2+y^2=1,z=1$.

[20M]

3) Find the characteristics pf the equation $pq=z$, and determine the integral surface which passes through the parabola $x=0,y=z.$

[20M]

4) Use Charpit’s method to find a complete integral to $p^2+q^2-2px-2qy+1=0.$

[20M]

5) Find the solution of the equation ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=e^{-1}cosy$
which $x\to 0$ as $x\to \mathrm{\infty }$ and has the value $cosy$ when $x=0$.

[20M]

6) One end of the string $(x=0)$ is fixed, and the point $x=a$ is made to oscillate, so that at time $t$ the displacement is $g(t)$.Show that the displacement $u(x,t)$ of the point $x$ at the time $t$ is given by $u(x,t)=f(ct-x)-f(ct+x)$ where $f$ is a function satisying the relation $f(t+2a)=f(t)-g({\displaystyle \frac{t+a}{c}}).$

[20M]

1998

1) Find the differential equation of the set of all right circular cones whose axes coincide with the Z-axis. Form the differential equation by eliminating $a$, $b$ and $c$ from $Z=a(x+y)+b(x-y)+abt+c$.

[20M]

2) Solve: $x{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}+y{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}+w{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }z}}=xyz$

[20M]

3) Find the integral surface of the linear partial differential equation $x(y^2+z){\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}}-y(x^2+z){\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}=(x^2-y^2)z$ through the straight line x+y=0,z=1.

[20M]

4) Use Charpit’s method to find complete integral of $2x[(z{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}{)}^2+1]=z){\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}}$.

[20M]

5) Find a real function V(x,y), which reduces to zero when $y=0$ and satisfies the equation ${\displaystyle \frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{y}^2}}=-4\pi (x^2+y^2)$.

[20M]

6) Apply Jacobi’s method to find a complete integral of the equation $2{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_1}}{x}_1{x}_3+3{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_2}}{x}_3^2+({\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_2}}{)}^2{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_3}}=0.$

[20M]

1997

1) Find the differential equation of all surfaces of revolution having z-axis as the axis of rotation.

[10M]

2) Form the differential equation by eliminating a and $b$ from $z=(x^2+a)(y^2+b)$.

[10M]

3) Find the equation of surfinces satisfying $4yzp+q+2y=0$ and passing through $y^2+z^2=1$, $x+z=2$

[20M]

4) Solve $(y+z)p+(z+x)q=x+y$.

[20M]

5) Use Charpit’s method to find complete integral of $z^2(p^2{z}^2+q^2)=1$.

[20M]

6) Solve $(D_X^3-D_Y^3)z=x^3{y}^3$.

[20M]

7) Apply Jacobi’s method to find complete integral of ${\mathrm{p}}_1{}^3{}^{+}{\mathrm{p}}_2{}^2+{\mathrm{p}}_3=1.$ Here $p_1={\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_1}},p_2={\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_2}},p_3={\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_3}}$ and $z$ is a function of ${\mathrm{x}}_{1,}{\mathrm{x}}_2,{\mathrm{x}}_3$

[20M]

1996

1) Find the differential equation of all spheres of radius $\lambda$ having their centre in xy-plane.

[10M]

2) Form differential equation by eliminating $f$ and $g$ from $z=f(x^2-y)+g(x^2+y)$

[10M]

3) Solve: $z^2(p^2+q^2+1)=c^2$.

[15M]

4) Find the integral surface of the equation $(x-y)y^{2}p+(y-x)x^{2}q=(x^2+y^2)z$ passing through the curve $\mathrm{x}\mathrm{z}={\mathrm{a}}^3,\mathrm{y}=0$

[15M]

5) Apply Charpit’s method to find the complete integral of $z=px+a+p^2+q^2$

[8M]

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

[15M]

7) Find a surface passing through the lines $z=x=0$ and $z-1=x-y=0$ satisfying ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}-4{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+4{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=0$

[15M]

1995

1) In the context of a partial differential equation of the first order in three independent variables, define and illustrate the terms: (i) the complete integral (ii) the singular integral

[15M]

2) Find the general integral of $(y+z+w){\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }x}}+(z+x+w){\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }y}}+(x+y+w){\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }z}}=x+y+z$

[15M]

3) Obtain the differential equation of the surfaces which are the envelopes of a one-parameter family of planes.

[15M]

4.(a) Explain in detail the Charpit’s method of solving the nonlinear partial differential equation $fx,y,z,({\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}},{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}})=0$

[10M]

4.(b) Solve ${\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_1}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_2}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_3}}=z^3{x}_1{x}_2{x}_3$

[10M]

5) Solve $(D_x^3-7D_x{D}_y^2-6D_y^3)z=\sin (x+2y)+e^{3x+y}$

[20M]