First Order but higher degree differential equation:
A differential equation involving first order but degree higher than one is called first order but higher degree differential equation.
Example:
Pn+A1Pn−1+A2Pn−2+…+An−aP+An
is a first-order and n-th degree differential equation, where p=dydx and A1,A2,…,An
are functions of x,y
Convenient of solution and various types of form, The equation is solved by five methods,
- Solvable for p
- Solvable for y
- solvable for x
- Clairauit's equation
- Lagrange's equation
Solvable for p:
If the equation
Pn+A1Pn−1+A2Pn−2+…+An−aP+An=0
can be expressed into n-factors of the first order and first degree
(p−α1)(p−α2)(p−α3)…(p−αn)=0
then equating to zero each factor, n-equation of the first order and first degree are obtained, Then the solutions are obtained by integration.
Question: Solve, p2+2xp−3x2=0
Solution: Given equation is,
p2+2xp−3x2=0or, p2+3xp−xp−3x2=0or, p(p+3x)−x(p+3x)=0or, (p−x)(p+3x)=0 ∴ p−x=0 p+3x=0
From (2) we get,
dydx−x=0dy−xdx=0
Integrating this we get,
y−x22−c2=0or, 2y−x2−c=0
From (3) we get,
dydx+3x=0or, dy+3x dx=0
Integrating this we get,
y+3x222−c2=02y+3x2−c=0
General solution of (1) is,
(2y−x2−c)(2y+3x2−c)=0
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