First Order ODE

First Order ODE

1. Separable?

   • No
     • Linear?
       • Yes
         • Put in the standard form dy/dx + P(x)y = Q(x)
           • Calculate the Integrating factor
             • Multiply the both sides of the equation by this factor
               • Rewrite the left-hand side of the equation; Integrate both sides
       • No
         • Homogeneous?
           • Yes
             • Put in the standard form dy/dx = F(y/x)
               • Use the substitutions: y=wx; dy/dx = w+ x(dw/dx)
                 • Now it's a separable equation!
           • No
             • Bernoulli?
               • Yes
                 • Put in the Standard Form: dy/dx + P(x)y = Q(x)^(n)
                   • Use the Special Substitution w= y^(1-n) to get linear
               • No
                 • Exact
                   • Yes
                     • Fxy = Fyx
                       • Integrate over x to find "Test Solution"; Take derivaive with respect to y
                   • No
                     • Try some substitution
   • Yes
     • Autonomous?
       • No
         • Integrate both sides
           • Look for singular solutions; Use the Initial Condition
       • Yes
         • The independent variable t do not appear explicitly
           • If f(x) = 0
             • The solutions are the Critical Points of the ODE

2. Check Before Attempting to Find a Solution

   • if f(x,y) is continuous
     • At Least One Solution
     • Besides that, if the partial derivative f/y is also continuous
       • Unique Solution

3. No

   • Reducible Second-Order?
     • Yes
       • General Form F(x,y, y', y") = 0
         • Missing y ( x,y', y")= 0
         • Missing x F(y, y', y")=0