273_pdfsam_math 54 differential equation solutions odd

273_pdfsam_math 54 differential equation solutions odd -...

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Exercises 5.2 Therefore, let’s try the substitution z ( t )= x ( t )+ y ( t ). We want a function z ( t ) that satisFes the two equations z 0 ( t ) z ( t 3 e 2 t and z 0 ( t )+2 z ( t )=3 e t , (5.9) simultaneously. We start by solving the Frst equation given in (5.9). This is a linear di±erential equation with constant coefficients which has the associated auxiliary equation r 1=0 . Hence, the solution to the corresponding homogeneous equation is z h ( t Ce t . By the method of undetermined coefficients, we see that a particular solution will have the form z p ( t Ae 2 t z 0 p = 2 Ae 2 t . Substituting these expressions into the Frst di±erential equation given in (5.9) yields z 0 p ( t ) z p ( t 2 Ae 2 t Ae 2 t = 3 Ae 2 t = 3 e 2 t A =1 . Thus, the Frst equation given in (5.9) has the general solution
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