2000

1) Solve: \(p q=x^{m} y^{n} z^{2 l}\)

[12M]

2) Prove that if \(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=1\) when \(z=0,\) the solution of the equation \(\left(\mathrm{S}-\mathrm{x}_{1}\right) \mathrm{p}_{1}+\left(\mathrm{S}-\mathrm{x}_{2}\right) \mathrm{p}_{2}+\left(\mathrm{S}-\mathrm{x}_{3}\right) \mathrm{p}_{3}=\mathrm{S}-\mathrm{z}\) can be given in the form \(S^{3}\left\{\left(x_{1}-z\right)^{3}+\left(x_{2}-z\right)^{3}+\left(x_{3}-z\right)^{3}\right\}^{4}=\left(x_{1}+x_{2}+x_{3}-3 z\right)^{3}\) where \(S=x_{1}+x_{2}+x_{3}+z\) and \(p_{i}=\dfrac{\partial z}{\partial x_{i}}, i=1,2,3\)

[12M]

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

[15M]

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

[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 \(k x(l-x),\) 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 \(\dfrac{\partial^2 z}{\partial x^2}+\dfrac{\partial^2 z}{\partial y^2}=e^{-1}cos y\)
which \(x\to 0\) as \(x\to \infty\) and has the value \(cos y\) 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 \(f(t+2a)=f(t)-g(\dfrac{t+a}{c}).\)

[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: \(x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}+w\dfrac{\partial u}{\partial z}=xyz\)

[20M]

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

[20M]

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

[20M]

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

[20M]

6) Apply Jacobi’s method to find a complete integral of the equation \(2\dfrac{\partial z}{\partial x_1}x_1x_3+3\dfrac{\partial z}{\partial x_2}x^2_3+(\dfrac{\partial z}{\partial x_2})^2\dfrac{\partial z}{\partial x_3}=0.\)

[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 \(z=\left(x^{2}+a\right)\left(y^{2}+b\right)\).

[10M]

3) Find the equation of surfinces satisfying \(4 y z p+q+2 y=0\) 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 \(z^{2}\left(p^{2} z^{2}+q^{2}\right)=1\).

[20M]

6) Solve \(\left(D_{X}^{3}-D_{Y}^{3}\right) z=x^{3} y^{3}\).

[20M]

7) Apply Jacobi’s method to find complete integral of \(\mathrm{p}_{1}{ }^{3}{ }^{+} \mathrm{p}_{2}{ }^{2}+\mathrm{p}_{3}=1 .\) Here \(p_{1}=\dfrac{\partial z}{\partial x_{1}}, p_{2}=\dfrac{\partial z}{\partial x_{2}}, p_{3}=\dfrac{\partial z}{\partial x_{3}}\) 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 \(z=f\left(x^{2}-y\right)+g\left(x^{2}+y\right)\)

[10M]

3) Solve: \(z^{2}\left(p^{2}+q^{2}+1\right)=c^{2}\).

[15M]

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

[15M]

5) Apply Charpit’s method to find the complete integral of \(z=p x+a+p^{2}+q^{2}\)

[8M]

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

[15M]

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

[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 \((y+z+w) \dfrac{\partial w}{\partial x}+(z+x+w) \dfrac{\partial w}{\partial y}+(x+y+w) \dfrac{\partial w}{\partial z}=x+y+z\)

[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 \(f x, y, z,\left(\dfrac{\partial z}{\partial x}, \dfrac{\partial z}{\partial y}\right)=0\)

[10M]

4.(b) Solve \(\dfrac{\partial z}{\partial x_{1}} \dfrac{\partial z}{\partial x_{2}} \dfrac{\partial z}{\partial x_{3}}=z^{3} x_{1} x_{2} x_{3}\)

[10M]

5) Solve \(\left( D _{ x }^{3}-7 D _{ x } D _{ y }^{2}-6 D _{ y }^{3}\right) z =\sin ( x +2 y )+ e ^{3 x + y }\)

[20M]