First Order PDEs
We will cover following topics
Types of first order PDEs
First order PDEs are classified into following four types:
(i) Linear: Linear in \(p\), \(q\) and \(z\), and coefficients of \(p\) and \(q\) are independent functions of \(x\) and \(y\).
It is of the form \(P(x, y) p+Q(x, y) q=R(x, y) z+S(x, y)\). For example, \(p+(x+y) q-z=e^{x}\).
(ii) Semilinear: Linear in \(p\) and \(q\), and and coefficients of \(p\) and \(q\) are independent functions of \(x\) and \(y\). It is of the form \(P(x, y) p+Q(x, y) q=R(x, y, z)\).
For example, \(p+(x+y) q-z^{3} x y\).
(iii) Quasilinear: Linear in \(p\) and \(q\). It is of the form \(P(x, y, z) p+Q(x, y, z) q=R(x, y, z)\).
For example, \(p z+(x+y) q-z^{3} x y=e^{x}\).
(iv) Nonlinear: PDEs which are not Linear. For example, \(p^{2}+q^{2}=1\).
Types of solutions of PDEs
(i) Complete Solution or Complete Integral: An expression of the type \(f(x, y, z, a, b)=0\) which is also a solution of the PDE.
For example, the expression \(f(x, y, z, a, b)=z-\left(a x+b y+a^{2}+b^{2}\right)\) is a solution to the PDE \(z-p x-q y-p^{2}-q^{2}=0\).
(ii) General Solution or General Integral: An expression of the form \(f(\phi, \psi)=0\) which is also a solution to the PDE.
For example, the expression \(f(\phi, \psi)=0\), where \(\phi=\dfrac{y}{x}\) and \(\psi=\dfrac{z}{x}\), is a solution to the PDE \(z-x p-y q\).
(iii) Singular Solution or Singular Integral: It is the envelope of the complete integral, if the envelope exists.
Quasilinear first order PDEs
The general solution of a quasilinear first order PDE \(a(x, y, u) u_{x}+b(x, y, u) u_{y}=c(x, y, u)\) is given by \(f(\phi, \psi)=0\), where \(f\) is an arbitrary function of \(\phi(x, y, u)\) and \(\psi(x, y, u)\), and \(\phi=c_{1}\), \(\psi=c_2\) are the solution curves of the characteristic equations
\[\dfrac{d x}{a}=\dfrac{d y}{b}=\dfrac{d z}{c}\]The solution curves defined by \(\phi(x, y, u)=c_{1}\) and \(\psi(x, y, u)=c_{2}\) are called the families of characteristic curves of the given quasilinear PDE.
Non-linear first order PDEs
Charpit’s method for nonlinear first order PDEs
Let \(f(x, y, z, p, q)=0\) be the given PDE.
We first write down the Charpit’s auxiliary equations, i.e.,
\(\dfrac{\mathrm{dp}}{\dfrac{\partial f}{\partial x}+p \dfrac{\partial f}{\partial z}}=\dfrac{\mathrm{dp}}{\dfrac{\partial f}{\partial y}+q \dfrac{\partial f}{\partial z}}=\dfrac{d z}{-p\left(\dfrac{\partial f}{\partial p}\right)-q\left(\dfrac{\partial f}{\partial q}\right)}=\dfrac{d x}{-\dfrac{\partial f}{\partial p}}=\dfrac{d y}{-\dfrac{\partial f}{\partial q}}=\dfrac{d F}{0}\),
and select any two partial fractions from these equations to find \(p\) and \(q\). We then write \(dz= pdx +qdy\) and integrate this to find the complete integral of the PDE.
Cauchy’s Method Of Characteristics
According to Cauchy’s method of characteristics, the characteristic curves of the quasilinear PDE \(a(x, y, z) \dfrac{\partial z}{\partial x}+b(x, y, z) \dfrac{\partial z}{\partial y}=c(x, y, z)\) are given by the equations:
\[\dfrac{d x}{a(x, y, z)}=\dfrac{d y}{b(x, y, z)}=\dfrac{d z}{c(x, y, z)}\]PYQs
Quasilinear first order PDEs
1) Find a complete integral of the partial differential equation \(2(p q+y p+q x)+x^{2}+y^{2}=0\).
[2017, 15M]
2) Final the general integral of the partial differential equation \((y+z x) p-(x+y z) q=x^{2}-y^{2}\).
[2016, 10M]
3) Solve the partial differential equation: \(\left(y^{2}+z^{2}-x^{2}\right) p-2 x y q+2 x z=0\), where \(p=\dfrac{\partial z}{\partial x}\) and \(q=\dfrac{\partial z}{\partial y}\) and \(q=\dfrac{\partial z}{\partial y}\).
[2015, 10M]
4) Solve for the general solution \(p \cos (x+y)+q \sin (x+y)=z\), where \(p=\dfrac{\partial z}{\partial x}\) and \(q=\dfrac{\partial z}{\partial y}\).
[2015, 15M]
5) Solve the PDE \((x+2 z) \dfrac{\partial z}{\partial x}+(4 z x-y) \dfrac{\partial z}{\partial y}=2 x^{2}+y\).
[2011, 12M]
6) Find the surface satisfying the PDE \(\left(D^{2}-2 D D^{\prime}+D^{\prime 2}\right) Z=0\) and the conditions that \(b Z=y^{2}\) when \(x=0\) and \(a Z=x^{2}\) when \(y=0\).
[2010, 12M]
7) Find complete and singular integrals of \(2 x z-p x^{2}-2 q x y+p q=0\) using Charpit’s method.
[2008, 15M]
8) Solve \(2 z x-p x^{2}-2 q x y+p q=0\).
[2007, 6M]
9) Solve: \(p x\left(z-2 y^{2}\right)=(z-q y)\left(z-y^{2}-2 x^{3}\right)\).
[2006, 12M]
10) Solve the equation \(p^{2} x+q^{2} y=z\), \(p=\dfrac{\partial z}{\partial x}\), \(q=\dfrac{\partial z}{\partial y}\).
[2006, 15M]
11) Obtain the general solution of \(\left(D-3 D^{\prime}-2\right)^{2} z=2 e^{2 x} \sin (y+3 x)\) where \(D=\dfrac{\partial}{\partial x}\) and \(D^{\prime}=\dfrac{\partial}{\partial y}\).
[2005, 30M]
12) Using Charpit’s method, find the complete solution of the partial differential equation \(p^{2} x+q^{2} y=z\).
[2004, 15M]
13) Find two complete integrals of the partial differential equation \(x^{2} p^{2}+y^{2} q^{2}-4=0\).
[2002, 12M]
14) Find the solution of the equation \(z=\dfrac{1}{2}\left(p^{2}+q^{2}\right)+(p-x)(q-y)\).
[2002, 12M]
15) Find the complete integral of the partial differential equation \(2 p^{2} q^{2}+3 x^{2} y^{2}=8 x^{2} q^{2}\left(x^{2}+y^{2}\right)\).
[2001, 12M]
16) Find the general integral of the equation \(\left\{m y(x+y)-n z^{2}\right\} \dfrac{\partial z}{\partial x}-\left\{l x(x+y)-n z^{2}\right\} \dfrac{\partial z}{\partial y}=(l x-m y) z\).
[2001, 12M]
Cauchy’s Method of Characteristics
1) Solve the first order quasilinear partial differential equation by the method of characteristics:
\(x\dfrac{\partial u}{\partial x}+(u-x-y)\dfrac{\partial u}{\partial y}=x+2y\) in \(x>0,-\infty<u<\infty\) with \(u=1+y\) on \(x=1\).
[2019, 15M]
2) Find the general solution of the partial differential equation
\((y^3x-2x^4)p+(2y^4-x^3y)q=9z(x^3-y^3)\),
where \(p=\dfrac{\partial z}{\partial x}\), \(q=\dfrac{\partial z}{\partial y}\), and its integral surface that passes through the curve: \(x=t\), \(y=t^2\), \(z=1\).
[2018, 15M]
3) Determine the characteristics of the equation \(z=p^{2}-q^{2}\) and find the integral surface which passes though the parabola \(4 z+x^{2}=0\).
[2016, 15M]
4) Solve the following partial differential equation, \(z p+y q=x\), \(x_{0}(s)=s\), \(y_{0}(s)=1\), \(z_{0}(s)=2 s\), by the method of characteristics.
[2010, 20M]
5) Find the characteristics of: \(y^{2} r-x^{2} t=0\) where \(r\) and \(t\) have their usual meanings.
[2009, 15M]
6) Find the general solution of the partial differential equation \((2 x y-1) p+\left(z-2 x^{2}\right) q=2(x-y z)\) and also find the particular solution which passes through the lines \(x=1\), \(y=0\).
[2008, 12M]
7) Find the particular integral of \(x(y-z) p+y(z-x) q=z(x-y)\) which represents a surface passing through \(x=y=z\).
[2005, 12M]
8) Find the complete integral of the partial differential equation \(\left(p^{2}+q^{2}\right) x=p z\) and deduce the solution which passes through the curve \(x=0\), \(z^{2}=4 y\).
[2004, 12M]
9) Find the characteristic strips of the equation \(x p+y q-p q=0\) and then find the equation of the integral surface through the curve \(z=\dfrac{x}{2}\), \(y=0\).
[2002, 20M]
10) Prove that for the equation \(z+p x+q y-1-p q x^{2} y^{2}=0\) the characteristic strips are given by \(x(t)=\dfrac{1}{B+C e^{-t}}\), \(y(t)=\dfrac{1}{A+D e^{-t}}\), \(z(t)=E-(A C+B D) e^{-t}\), \(p(t)=A\left(B+C e^{-t}\right)^{2}\), \(q(t)=B\left(A+D e^{-t}\right)^{2}\) where \(A\), \(B\), \(C\), \(D\) and \(E\) are arbitrary constants. Hence find the values of these arbitrary constants if the integral surface passes the line \(z=0\), \(x=y\).
[2001, 30M]