First Order ODE

First Order ODE

1. Separable?

   • Yeah!
     • Autonomous?
       • No
         • Integrate Both Sides
           • Look For Singular Solutions. Use The IInitial Condition
       • Yes
   • No
     • Linear?
       • Yes
         • Put in the Standad form: dy/dx + P(x)y = Q(x)
           • Identify P(x)
             • Calculate the integrating factor and multiply the both sides of the equation by it.
               • Rewrite the left-hand side of the eq. as the derivative of the product
                 • Integrate Both sides.
       • No
         • Homogenous?
           • Yes
             • Put in the Standard form: dy/dx = F(y/x)
               • Try the substitutions y= wx and dy/dx = w + x(dw/dx)
                 • Now it is a separable equation
           • No
             • Bernoulli?
               • Yes
                 • Put in the Standard form: dy/dx + P(x)y = Q(x)y^n
                   • Use the special substitution w= y^(1-n)
               • No
                 • • Yes
                   • No
                     • Try sSme Substitution

2. CHECK BEFORE ATTEMPT!!!

   • if f(x,y) is continuous,
     • At least one solution.
     • Besides that, if the partial derivative f/y is continuous
       • Unique solution

3. Separable?

   • No
     • Linear?
       • Yes
       • No
         • Homogeneous?
           • Yes
           • No
             • Bernoulli?
               • Yes
               • No
                 • • Yes
                   • No
                     • Try some substitution
   • Yes
     • Autonomous?
       • No
         • Integrate both sides
           • Look for singular solutions; Use the Initial Condition
       • Yes

4. 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