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Types of Solutions in Partial Differential Equation | Study With Nitin

In Partial Differential Equation questions are solving to some important forms Lagrange's Linear Equation, Charpit's Method and etc. Here the concept of solutions are very vital to understand for solving the P.D Equation.
Types of Solutions in Partial Differential Equation

Types of Solutions

There are four types of solutions are listed below -
  • Complete Solution
  • Particular Solution
  • Singular Solution
  • General Solution

Complete Solution

When a partial differential equation f(x,y,z,p,q) = 0     ...(i)
where, z is the function of independent variables x and y and f  be the function of dependent variables.

Here we obtain a relation
g(x,t,z,a,b) = 0
there are many arbitary constants as a number of independent variables, then g(x,y,z,p,q) = 0
is called the complete solution or integral of eq(i)
for example,
z = ax + by +f(a,b) is the complete solution of
z = px +qy +f(p,q)

Particular Solution

First of all we have a differential equation 
f(x,y,z,p,q) = 0     ...(i)
Here, a solution obtainted form complete solution by giving some particular values of arbitrary constant is called particular solution of eq.(i)
For Example: z = x + 2y + 2 is a particular solution of
                       z = px + py + pq

Singular Solution

The equation of envelope of the surfaces represented by complete solution of the given P.D equation.

Now, if a P.D is

f(x,y,z,p,q) = 0     ...(i)

Then the complete solution will be 
g(x,y,z,a,b) = 0     ...(ii)

Now eliminating a,b form trhe equation (ii), to obtained the relation of x,y,z we have
g(x,yx,z,a,b) = 0, del(z)/del(a) = 0, del(z)/del(b) = 0 is called the singular solution of (i), if it satisfies the equation(i).

General Solution

We know that,
the complete solution if g(x,y,z,a,b) = 0 of  f(x,y,z,p,q) = 0
Here, if b = ∅(a)
Then, g(x,y,z,a,∅(a)) = 0  is one parameter family of surfaces f(x,y,z,p,q) = 0 
If we eliminating the arbitrary constant a form eq. g(x,y,z,a,∅(a)) = 0 and del(g)/del(a) = 0 (to obtained the relation between  x,y,z) is the general solution of (i) if it satisfies the equation.

FAQs for Types of Solutions in Partial Differential Equation

Q 1. What is the full form of PDE?
Ans. Partial Differential Equation.

Q 2. How many solutions are in PDE?
Ans. There are four types of solution in PDE.

Q. 3. How many solutions does a partial differential equation have?
Ans. Either has one or infinitely many solution.

Q 4. What is the type of a solution in differential equation?
Ans. There are four types of solution are in PDE i.e., General solution, particular solution, singular solution, complete solution.

Q 5. What are the general solution of PDE?
Ans. A family of solution that includes all possible solution to differential eq. 

Q 6. How many solutions are there to a differential equation?
Ans. Four.

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