Solve the following initial value problems using method of undetermined coefficients: (i) y''−9y = ex+x−1, y(0) = −1,y'(0) = 1,
Correponding homogeneous differential equation
Characteristic (auxiliary) equation
"r_1=3, r_2=-3"
The general solution of the homogeneous differential equation is
Find the particular solution of the non homogeneous differential equation
"y_p'=Ae^x+B"
"y_p''=Ae^x"
Substitute
"A=-1\/8"
"B=-1\/9"
"C=1\/9"
The general solution of the non homogeneous differential equation is
"y'(0) = \u22121,y(0) = 1"
"c_1+c_2-\\dfrac{1}{8}+\\dfrac{1}{9}=1"
"c_2=\\dfrac{73}{72}-c_1"
"6c_1-\\dfrac{219}{72}=-\\dfrac{55}{72}"
"c_1=\\dfrac{82}{216}"
The solution of the given IVP is
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