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Second Order Linear Differential Equation (Non-constant coefficients)

  • #1
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Homework Statement


Find the general solution for the following differential equation:
y'' + x(y')^2 = 0.


Homework Equations


Integration, differentiation...


The Attempt at a Solution


Usually for these sort of DE you could use the substitution v(x) = y'(x) and this would simplify such an equation to a first order DE. The (y')^2 part is throwing me off as this would give the equation:
v(x)' + x(v(x))^2 = 0.
This would be grand if it weren't for the v(x)^2 term. Any ideas on how to get around this? Thanks.
 

Answers and Replies

  • #2
CAF123
Gold Member
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Are you worried about the v^2 term because it makes the eqn non-linear? If so, there are ways to solve such an equation.
 
  • #3
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Yeah. We've only covered linear DE's.
 
  • #4
CAF123
Gold Member
2,906
88
Yeah. We've only covered linear DE's.
Are you at all familiar with separation of variables?
 
  • #5
310
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Yeah. So I'll have v'(x) = (-x)(v(x))^2 and I can then separate the variables and solve. Thanks!
 

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