We will cover following topics

Types of first order PDEs
Types of solutions of PDEs
Quasilinear first order PDEs
Non-linear first order PDEs
- Charpit’s method for nonlinear first order PDEs
Cauchy’s Method Of Characteristics
PYQs
- Quasilinear first order PDEs
- Cauchy’s Method of Characteristics

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 $f (x, y, z, a, b) = z - (a x + b y + a^{2} + b^{2})$ 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 $f (ϕ, ψ) = 0$ which is also a solution to the PDE.

For example, the expression $f (ϕ, ψ) = 0$ , where $ϕ = \frac{y}{x}$ and $ψ = \frac{z}{x}$ , 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 $f (ϕ, ψ) = 0$ , where $f$ is an arbitrary function of $ϕ (x, y, u)$ and $ψ (x, y, u)$ , and $ϕ = c_{1}$ , $ψ = c_{2}$ are the solution curves of the characteristic equations

\frac{d x}{a} = \frac{d y}{b} = \frac{d z}{c}

The solution curves defined by $ϕ (x, y, u) = c_{1}$ and $ψ (x, y, u) = c_{2}$ 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.,

$\frac{d p}{\frac{\partial f}{\partial x} + p \frac{\partial f}{\partial z}} = \frac{d p}{\frac{\partial f}{\partial y} + q \frac{\partial f}{\partial z}} = \frac{d z}{- p (\frac{\partial f}{\partial p}) - q (\frac{\partial f}{\partial q})} = \frac{d x}{- \frac{\partial f}{\partial p}} = \frac{d y}{- \frac{\partial f}{\partial q}} = \frac{d F}{0}$ ,

and select any two partial fractions from these equations to find $p$ and $q$ . We then write $d z = p d x + q d y$ 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 $a (x, y, z) \frac{\partial z}{\partial x} + b (x, y, z) \frac{\partial z}{\partial y} = c (x, y, z)$ are given by the equations:

\frac{d x}{a (x, y, z)} = \frac{d y}{b (x, y, z)} = \frac{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: $(y^{2} + z^{2} - x^{2}) p - 2 x y q + 2 x z = 0$ , where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$ and $q = \frac{\partial z}{\partial y}$ .

[2015, 10M]

4) Solve for the general solution $p \cos (x + y) + q \sin (x + y) = z$ , where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$ .

[2015, 15M]

5) Solve the PDE $(x + 2 z) \frac{\partial z}{\partial x} + (4 z x - y) \frac{\partial z}{\partial y} = 2 x^{2} + y$ .

[2011, 12M]

6) Find the surface satisfying the PDE $(D^{2} - 2 D D^{'} + D^{' 2}) Z = 0$ 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: $p x (z - 2 y^{2}) = (z - q y) (z - y^{2} - 2 x^{3})$ .

[2006, 12M]

10) Solve the equation $p^{2} x + q^{2} y = z$ , $p = \frac{\partial z}{\partial x}$ , $q = \frac{\partial z}{\partial y}$ .

[2006, 15M]

11) Obtain the general solution of ${(D - 3 D^{'} - 2)}^{2} z = 2 e^{2 x} \sin (y + 3 x)$ where $D = \frac{\partial}{\partial x}$ and $D^{'} = \frac{\partial}{\partial y}$ .

[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 $z = \frac{1}{2} (p^{2} + q^{2}) + (p - x) (q - y)$ .

[2002, 12M]

15) Find the complete integral of the partial differential equation $2 p^{2} q^{2} + 3 x^{2} y^{2} = 8 x^{2} q^{2} (x^{2} + y^{2})$ .

[2001, 12M]

16) Find the general integral of the equation ${m y (x + y) - n z^{2}} \frac{\partial z}{\partial x} - {l x (x + y) - n z^{2}} \frac{\partial z}{\partial y} = (l x - m y) z$ .

[2001, 12M]

Cauchy’s Method of Characteristics

1) Solve the first order quasilinear partial differential equation by the method of characteristics:

$x \frac{\partial u}{\partial x} + (u - x - y) \frac{\partial u}{\partial y} = x + 2 y$ 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

$(y^{3} x - 2 x^{4}) p + (2 y^{4} - x^{3} y) q = 9 z (x^{3} - y^{3})$ ,

where $p = \frac{\partial z}{\partial x}$ , $q = \frac{\partial z}{\partial y}$ , 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 $(2 x y - 1) p + (z - 2 x^{2}) q = 2 (x - y z)$ 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 $(p^{2} + q^{2}) x = p z$ 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 $z = \frac{x}{2}$ , $y = 0$ .

[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 $x (t) = \frac{1}{B + C e^{- t}}$ , $y (t) = \frac{1}{A + D e^{- t}}$ , $z (t) = E - (A C + B D) e^{- t}$ , $p (t) = A {(B + C e^{- t})}^{2}$ , $q (t) = B {(A + D e^{- t})}^{2}$ 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]

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