There are five types of methods for solving first order non-linear partial differential equations: I) Equations containing only p and q variables. II) Equations relating z as a function of u. III) Equations that can be separated into functions of single variables. IV) Clairaut's Form where the solution is directly substituted. V) Charpit's Method which is a general method taking integrals of auxiliary equations to solve dz=pdx+qdy and find the solution. These types cover a range of applications including Poisson's, Helmholtz's, and Schrödinger's equations in fields like electrostatics, elasticity, wave theory and quantum mechanics.

1. There are five ways of non-linear partial differential equations
of first order and their method of solution as given below.
•Type I:
•Type II:
•Type III: (variable separable method)
•Type IV: Clairaut’s Form
•CHARPIT’S METHOD

2. Type I: f(p, q)=0
Equations of the type f(p, q)=0 i.e. equations containing p and q only
Let the required solution be
and
Substituting these values in , f(p, q)=0 we get f(a, b)=0
From this, we can obtain b in terms of a (or) a in terms of b
Let , then the required solution is

3. Type II:
Let us consider the Equations of the type
Let z is a function of u ie. z = and
Now,
is the 1st order differential equation in
terms of dependent variable z and independent variable u.
Solving this differential equation and finally substitute
gives the required solution.
u
z
a
y
u
u
z
y
z
q
∂
∂
=
∂
∂
∂
∂
=
∂
∂
=

4. Type III – f1(p,x) = f2(q,y) (variable separable method)
Let us consider differential equation of the form
f1(p,x) = f2(y,q)
Let f1(p,x) = f2(y,q) = a (say)
Now f1(p,x) = a → p = Ψ1(x) (ie. Writing p in terms of x)
f2(q,y) = a→ q = Ψ2(y) (ie. Writing q in terms of y)
Now,
dz =
= pdx + qdy
so dz = Ψ1(x)dx + Ψ2(y)dy
dy
y
z
dx
x
z
∂
∂
+
∂
∂

5. Type IV: Clairaut’s Form
Equations of the form
Let the required solution be
then
Required solution is
i.e. Directly substitute a in place of p and b in place of q in the
given equation.

6. CHARPIT’S METHOD
This is a general method to find the complete integral of the non-
linear PDE of the form f (x , y, z, p, q) = 0
Now Auxillary Equations are given by
Here we have to take the terms whose integrals are easily
calculated, so that it may be easier to solve and finally substitute in
the equation dz = pdx + qdy
Integrate it, we get the required solution.
Note :that all the above (TYPES) problems can be solved in this
method.

7. Applications of PDE
• Poisson’s Equation
which arises in electrostatics, elasticity theory and elsewhere.
• Helmholtz's Equation
which arisis in wave theory.
• Schrödinger's Equation
which arises in quantum mechanics.

8. THANK YOU