Solution by Substitution Homogeneous Differential Equations Bernoulli’s Equation Reduction to Separation of Variables Conclusion
                                MATH312
               Section 2.5: Solutions by Substitution
                           Prof. Jonathan Duncan
                               Walla Walla College
                            Spring Quarter, 2007
  Solution by Substitution Homogeneous Differential Equations Bernoulli’s Equation Reduction to Separation of Variables Conclusion
  Outline
       1  Solution by Substitution
       2  Homogeneous Differential Equations
       3  Bernoulli’s Equation
       4  Reduction to Separation of Variables
       5  Conclusion
  Solution by Substitution Homogeneous Differential Equations Bernoulli’s Equation Reduction to Separation of Variables Conclusion
  A Motivating Example
       In this last section of chapter 2, we introduce no new methods of
       solving DEs but rather look at ways to reduce a DE to a type we
       already know how to solve.
       Example
       Solve the following differential equation.
                           (y2 +yx) dx +x2 dy = 0
       Your first impulse might be to try exact solution methods.
       However:
            The equation is not exact.
            M −N      2y−x      N −M      x−2y
             y   x =       and x     y =       .
              N        x2         M       y2+yx
            Finally, dy = −y2+yx is neither separable nor linear.
                    dx       x2
  Solution by Substitution Homogeneous Differential Equations Bernoulli’s Equation Reduction to Separation of Variables Conclusion
  What is a Homogeneous DE? (this time...)
       Unfortunately, the name for differential equations in which our first
       substitution works has already been used in this class.
       Definition
       If f (x, y) is a function such that f (tx,ty) = tαf (x,y) for some
       real number α, then f is a homogeneous function of degree α.
       Definition
       If M(x,y) dx +N(x,y) dy = 0 is a first order differential equation
       in differential form, then it is called homogeneous if both M and
       N are homogeneous functions of the same degree.