Early Diff EQ Solving Methods (Chp 1 status yo)

[Kevin2341]

Alright, got a quick differential equation question.

So far in my DE class, we have learned 3 methods of solving ODE's
-Separation of Variables
-"Guessing" (the y=yh+yp method)
-Integrating Factors

How do you know when to use these methods and when not to? I understand the methods to use each of these three (Although guessing is still pretty sketchy when it comes to actually guessing).

Does it have something to do with linearity? Or is it simply just which ever method is "nicer"? I know with integrating factors sometimes you end up with integrals that are impossible to solve, so in that case, you'd need to resort to guessing.

 

[LCKurtz]

Alright, got a quick differential equation question.

So far in my DE class, we have learned 3 methods of solving ODE's
-Separation of Variables
-"Guessing" (the y=yh+yp method)
-Integrating Factors

How do you know when to use these methods and when not to? I understand the methods to use each of these three (Although guessing is still pretty sketchy when it comes to actually guessing).

Does it have something to do with linearity? Or is it simply just which ever method is "nicer"? I know with integrating factors sometimes you end up with integrals that are impossible to solve, so in that case, you'd need to resort to guessing.

I would look at the problem and ask myself, in this order:

1. Is if first order? If so then
2. Is it constant coefficient? If so use the characteristic equation and $y_h,\, y_p$ if it is non-homogeneous.
3. Is it linear? If so solve by integrating factor.
4. Is it separable? If so separate and integrate.
5. Write it as M(x,y)dx + N(x,y)dy = 0. Is it exact? Homogeneous (y = ux substitution)? Is there an integrating factor function of x or function of y?
6. Is it some special equation like a Bernoulli nonlinear equation?

Now if it is second order then:
7. Is it constant coefficient? f so use the characteristic equation and $y_h,\, y_p$ if it is non-homogeneous.
8. Is the $y'$ term missing? If so let $u=y'$ making a first order equation in $y'$.
9. Do you know one solution so you can reduce the order?
10. Is it linear, maybe solvable by series?

That should get you started. You usually won't have to go all the way down the list.