Partial Differential Equations

Module I: Partial Differential Equations – All Important Formulas

1. General First-Order PDE

F\bigl(x,y,u,p,q\bigr)=0

where

p=\frac{\partial u}{\partial x},\qquad q=\frac{\partial u}{\partial y}.

2. Linear First-Order PDE (Lagrange’s Form)

P(x,y,u)\,p + Q(x,y,u)\,q = R(x,y,u)

Subsidiary (Characteristic) Equations

\frac{dx}{P}=\frac{dy}{Q}=\frac{du}{R}

General Solution
If two independent integrals are

u_1(x,y,u)=c_1,\qquad u_2(x,y,u)=c_2,

then the general solution is

\Phi(u_1,u_2)=0 \quad\text{or}\quad u_1=f(u_2).

3. Non-Linear First-Order PDE

F(x,y,u,p,q)=0

Charpit’s Auxiliary Equations

\frac{dx}{\dfrac{\partial F}{\partial p}} =\frac{dy}{\dfrac{\partial F}{\partial q}} =\frac{du}{p\dfrac{\partial F}{\partial p}+q\dfrac{\partial F}{\partial q}} =\frac{-dp}{\dfrac{\partial F}{\partial x}+p\dfrac{\partial F}{\partial u}} =\frac{-dq}{\dfrac{\partial F}{\partial y}+q\dfrac{\partial F}{\partial u}}

From these we obtain a relation

p=p(x,y,u,a),\qquad q=q(x,y,u,a)

which is substituted back into $F=0$ and then integrated.

4. Cauchy’s Method of Characteristics

The characteristic strips satisfy the system of ODEs

\begin{aligned} \frac{d x}{d s} & = \frac{\partial F}{\partial p}, & \frac{d y}{d s} & = \frac{\partial F}{\partial q}, & \frac{d u}{d s} & = p \frac{\partial F}{\partial p} + q \frac{\partial F}{\partial q}, \\ \frac{d p}{d s} & = - (\frac{\partial F}{\partial x} + p \frac{\partial F}{\partial u} \end{aligned}

Initial curve $\Gamma$ : $x=x_0(t)$ , $y=y_0(t)$ , with compatible , .

5. Linear PDE of Higher Order with Constant Coefficients

\sum_{r=0}^{n}a_r\frac{\partial^n u}{\partial x^{n-r}\partial y^r}=f(x,y) \qquad\bigl(a_r=\text{constants}\bigr)

Symbolic Form

F(D,D')u=f(x,y),\qquad D=\frac{\partial}{\partial x},\quad D'=\frac{\partial}{\partial y}.

Homogeneous Equation ( $f=0$ )

Assume $u=e^{mx+ny}$ . The auxiliary equation is

F(m,n)=0.

If roots are $(m_i,n_i)$ , particular solutions are of the form $e^{m_ix+n_iy}$ .

General Solution of Homogeneous Equation

u_c=\sum_i\phi_i(y+m_ix) \quad\text{(or appropriate forms for complex/repeated roots)}.

Particular Integral

u_p=\frac{1}{F(D,D')}f(x,y)

Useful rules:

$f=e^{ax+by}$ : $\displaystyle\frac{1}{F(a,b)}e^{ax+by}$ (if )

6. Equations Reducible to Linear PDEs with Constant Coefficients

Type 1 – Change of Independent Variables
If the equation is of the form

f\Bigl(x\frac{\partial}{\partial x},y\frac{\partial}{\partial y}\Bigr)u=g(x,y),

put

x=e^X,\qquad y=e^Y \quad\Rightarrow\quad x\frac{\partial}{\partial x}=\frac{\partial}{\partial X},\quad y\frac{\partial}{\partial y}=\frac{\partial}{\partial Y}.

Type 2 – Euler–Cauchy Type (Homogeneous)

\Bigl(a_0x^n\frac{\partial^n}{\partial x^n}+a_1x^{n-1}y\frac{\partial^n}{\partial x^{n-1}\partial y}+\cdots+a_ny^n\frac{\partial^n}{\partial y^n}\Bigr)u=f(x,y)

is reduced by the same substitution $x=e^X$ , $y=e^Y$ to a constant-coefficient equation.

Type 3 – Other Transformations
Any linear PDE that can be transformed (by a suitable change of independent or dependent variables) into an equation with constant coefficients is solved by the methods of §5 after the transformation.

These are the complete standard formulas required for Module I.