′ ( 0 )=8? ? The initial conditions lead to the system of equations which is equivalent to The solution to the IVP is given by . ______________________________________________________________________ 4. (8 points) The following differential equation may be solved by either performing a substitution to reduce it to a separable equation or by performing a different substitution to reduce it to a homogeneous equation. Display the substitution to use and perform the reduction, but do not attempt to solve the separable or homogeneous equation you obtain . The key to this puzzle is the solution to the linear system which is equivalent to , geometrically, a pair of parallel lines.Consequently, a suitable substitution is given by z = x -2 y . After a little routine algebra, the reduction results in the separable DE ______________________________________________________________________ 10 Point Bonus: Although you will later learn a body of theory that will make the solution of the following following 2rd order equation routine,
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