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Is The Equation First Order? - Coggle Diagram

- - - - Constant Coefficients?
        
        Trial solution of y = e^mx to form auxiliary equation and solve :checkered_flag:
        
        Is it Euler-Cauchy?
        
        Does it have polynomial coefficients?
        
        Is x=0 an ordinary point of the equation?
        
        Use Leibniz-McLaurin power series method {3.3.4-18} :checkered_flag:
        
        Substitute: x=x0 + z and then apply Leibniz-McLaurin :checkered_flag:
        
        NA
        
        Trial solution of y = x^m to form Auxiliary equation ...
      - Does it have constant coefficients AND is the free term quasi-polynomial?
        
        Can you solve the corresponding homogeneous equation?
        
        Can you find at least one non-trivial solution to the homogeneous equation?
        
        No need for understanding
        
        Reduce the order of the equation using the substitution: g(x) = v(x)y1(x) and the equation becomes first order linear. :checkered_flag:
        
        Take linearly independent solutions y1 and y0 and use the method of variation of parameters :checkered_flag:
        
        Solve using the method of undetermined coefficients :checkered_flag:
    - - Does the equation involve y' and y'' but not y?
        
        Try ad hoc substitution using (in)dependant variable to reduce the order to linear first order ODE and if it doesn't work then its beyond the scope of the course:checkered_flag:
        
        Reduce the order using the substitution: y'(x) = z(x) :checkered_flag:
      - Reduce the order with the substitution: y' = p(y) {3.1.2} :checkered_flag: