Introduction
We will cover following topics
Family Of Surfaces In Three Dimensions
Consider the family of suraces given by:
\(\qquad S_{\lambda} : z=G(x, y ; \lambda) \qquad (1)\),
where \(\lambda\) is the parameter.
Keeping \(x,y,z\) fixed, differentiate \(z=G(x, y ; \lambda)\) with respect to \(\lambda\) to get:
\(\qquad 0=G_{\lambda}(x, y ; \lambda) \qquad (2)\),
Suppose \(C_{\lambda}\) is the curve of intersection given by (1) and (2). The envelope of the family \(S_{\lambda}\) is the union of \(C_{\lambda}\) for all \(\lambda\).
If we can solve for \(\lambda\) from (2) in terms of \(x\) and \(y\) as \(\lambda = g(x,y)\), then the envelope is analytically represented by the equation \(z=G(x, y, g(x, y))\).
Formulation Of Partial Differential Equations
PDEs can be formed by the elimination of arbitrary constants or arbitrary functions.
Elimination of arbitrary constants
In these types of problems, we write \(z=f(x,y,a,b)\) and partially differentiate it with respect to \(x\) and \(y\) respectively. Then, we eliminate the arbitrary constants from the two equations.
The technique can be similarly extended to functions of more than two variables.
Elimination of arbitrary constants
In these types of problems, if we are asked to remove a single arbitrary function, we take partial derivatives with respect to \(x\) and \(y\).
Similarly, if we are asked to remove more than one arbitrary function, we take higher order partial derivatives and from these equations, we remove the arbitrary functions.
PYQs
Family Of Surfaces In Three Dimensions
1) Form a partial differential equation of the family of surfaces given by the following expression:
\(\psi(x^2+y^2+2z^2,y^2-2zx)=0\).
[2019, 10M]
2) Find the partial differential equation of the family of all tangent planes to the ellipsoid: \(x^2+4y^2+4z^2=4\), which are not prependicular the the \(xy\) plane.
[2018, 10M]
3) Find the general equation of surfaces orthogonal to the family of spheres given by \(x^{2}+y^{2}+z^{2}=c z\).
[2016, 10M]
4) Find the surface which intersects the surfaces of the system \(z(x+y)=C(3 z+1)\), (\(C\) being a constant) orthogonally and which passes through the circle \(x^{2}+y^{2}=1\), \(z=1\).
[2013, 15M]
5) Find the surface satisfying \(\dfrac{\partial^{2} z}{\partial x^{2}}=6 x+2\) and touching \(z=x^{3}+y^{3}\) along its section by the plane \(x+y+1=0\).
[2011, 20M]
6) Show that the differential equation of all cones which have their vertex at the origin is \(p x+q y=z\). Verify that this equation is satisfied by the surface \(y z+z x+x y=0\).
[2009, 12M]
7) Find the integral surface of: \(x^{2} p+y^{2} p+z^{2}=0\) which passes through the curve: \(x y=x+y\), \(z=1\).
[2009, 6M]
8) Transform the equation \(y z_{x}-x z_{y}=0\) into one in polar coordinates and thereby show that the the the solution of the given equation represents surfaces of revolution.
[2007, 12M]
9) Find the surface passing through the parabolas \(z=0\), \(y^{2}=4 a x\) and \(z=1\), \(y^{2}=-4 a x\) and satisfying the equation \(x \dfrac{\partial^{2} z}{\partial x^{2}}+2 \dfrac{\partial z}{\partial x}=0\).
[2006, 15M]
10) Formulate partial differential equation for surfaces whose tangent planes form a tetrahedron of constant volume with the coordinate planes.
[2005, 12M]
11) Find the integral surface of the following partial differential equation:
\(x\left(y^{2}+z\right) p-y\left(x^{2}+z\right) q=\left(x^{2}-y^{2}\right) z\)
[2004, 12M]
12) Show that the differential equations of all cones which have their vertex at the origin are \(p x+q y=z\). Verify that \(y z+z x+x y=0\) is a surface satisfying the above equation.
[2003, 12M]
13) Solve \(\dfrac{\partial^{2} z}{\partial x^{2}}-\dfrac{\partial^{2} z}{\partial y^{2}}-3 \dfrac{\partial z}{\partial x}+3 \dfrac{\partial z}{\partial y}=x y+e^{x+2 y}\).
[2003, 15M]
14) Solve the equation \(p^{2}-q^{2}-2 p x-2 q y+2 x y=0\) using Charpit’s method. Also find the singular solution of the equation, if it exists.
[2003, 15M]
15) Write down the system of equations for obtaining the general equation of surfaces orthogonal to the family given by \(x\left(x^{2}+y^{2}+z^{2}\right)=C_{1} y^{2}\).
[2001, 10M]
Formulation Of Partial Differential Equations
1) Form a partial differential equation by eliminating the arbitrary functions \(f\) and \(g\) from \(z=y f(x)+x g(y)\).
[2013, 10M]
2) Solve the partial differential equation \(\left(D-2 D^{\prime}\right)\left(D-D^{\prime}\right)^{2} z=e^{x+y}\).
[2012, 12M]
3) Solve the partial differential equation \(p x+q y=3 z\).
[2012, 20M]
4) Form the partial differential equation by elimination the arbitrary function \(f\) given by: \(f\left(x^{2}+y^{2}, z-x y\right)=0\).
[2009, 6M]
5) Form a partial differential equation by eliminating the function \(f\) from:
\(z=y^{2}+2 f\left(\dfrac{1}{x}+\log y\right)\).
[2007, 6M]
6) Frame the partial differential equation by eliminating the arbitrary constants \(a\) and \(b\) from \(\log (a z-1)=x+a y+b\).
[2002, 10M]