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MATH 129 lesson 10-12

 



    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+A1Pn1+A2Pn2++AnaP+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,

  1. Solvable for p
  2. Solvable for y 
  3. solvable for x
  4. Clairauit's equation
  5. Lagrange's equation

    Solvable for p:
If the equation Pn+A1Pn1+A2Pn2++AnaP+An=0 can be expressed into n-factors of the first order and first degree (pα1)(pα2)(pα3)(pαn)=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+2xp3x2=0
    Solution: Given equation is, 
p2+2xp3x2=0or, p2+3xpxp3x2=0or, p(p+3x)x(p+3x)=0or, (px)(p+3x)=0  px=0 p+3x=0
From (2) we get, dydxx=0dyxdx=0
Integrating this we get, yx22c2=0or, 2yx2c=0
From (3) we get, dydx+3x=0or, dy+3x dx=0
Integrating this we get, y+3x222c2=02y+3x2c=0
General solution of (1) is, (2yx2c)(2y+3x2c)=0


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