PDE

z is a function of x,y

$$ \begin{align*} p=\frac{\partial z}{\partial x}\\ q=\frac{\partial z}{\partial y}\\ r=\frac{\partial^2 z}{\partial x^2}\\ s=\frac{\partial^2 z}{\partial x\partial y}\\ t=\frac{\partial^2 z}{\partial y^2} \end{align*} $$

f(x,y,z,p,q,r,s,t)=0 is a Partial differential equation

Order: It is the order of the highest partial derivative appearing in differential equation.

Formation of PDE #

arbitrary constants #

f(x,y,z,a,b)=0 We form a PDE by eliminating the arbitrary constants a,b.

arbitrary function #

We take the derivative of a relation to obtain a PDE

F(u,v)=0; u(x,y,z), v(x,y,z)

taking derivative wrt x,y:

$$ \begin{align*} F_u(u_x+u_zz_x)+F_v(v_x+v_zz_x)=0\\ F_u(u_y+u_zz_y)+F_v(v_y+v_zz_y)=0\\ \begin{vmatrix}u_x+u_zz_x & v_x+v_zz_x\\ u_y+y_zz_y & v_y+v_zz_y\end{vmatrix}=0\\ \\ \text{We get the form: }Pp+Qq=R\\ P=\frac{\partial (u,v)}{\partial (y,z)}\\ Q=\frac{\partial (u,v)}{\partial (z,x)}\\ R=\frac{\partial (u,v)}{\partial (x,y)} \end{align*} $$

Classification of a PDE #

1. Linear
   1. P(x,y)p+Q(x,y)q=R(x,y)z+S(x,y)
2. Semi-linear
   1. P(x,y)p+Q(x,y)q=R(x,y,z)
3. Quasi-linear
   1. P(x,y,z)p+Q(x,y,z)q=R(x,y,z) (called Lagrange’s linear equation)
4. Non-linear: none of the above

Solutions of a PDE #

• Complete solution
  • Function of the form F(x,y,z,a,b)=0 where a,b are arbitrary constants
• General solution
  • Function of the form F(u,v)=0 where F is an arbitrary function of u,v satisfying PDE is called general solution.