Page 291 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes So we have Clairaut equation type
p = a, q = b,

a 4 b 4
so z = ax + by + ... (22)
ab
is the complete solution.

Self Assessment

7. Find the complete integral of
4
4
2 2
z = px + qy + p + q + p q
8. Find the solution of
2
p (q + 1) = q (z b)
18.4 Summary

 Charpit method is quite useful in finding the complete integral of the first order partial
differential equation.
 Here we are interested in setting up auxiliary equations with the help of which the values
of p and q are obtained.

z z
 Knowledge of the first derivatives , or p and q respectively help in finding the
x y
complete integral involving two arbitrary constants.
18.5 Keywords

Charpit s method helps in finding the complete integral of the first order partial differential
equation.

Jacobi s method: It deals with two independent variables and so to solve partial differential
equation having more than two independent variables we have to take the help of Jacobi s
method.

18.6 Review Questions

Solve by Charpit s method:
2
2
1. p x + q y = z
2
2
2
2. p y q = y x 2
3. yp = 2yx + log q
2
2 2
2
4. z (p z + q ) = 1
Answers: Self Assessment
y
1. z = c x + c e (y + c ) c
1 2 1 1
2
2. z = ax + 3a y + b

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