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Solve the initial value problem y''+3y'+2y = 3e[tex]^{2t}[/tex]+1 with initial values y(0) = 1, y'(0) = 1.

I am unsure if I am going about the solution correctly.

1.) Find the characteristic equation.

r[tex]^{2}[/tex]+3r+2=0 [tex]\Rightarrow[/tex] (r + 1)(r + 2) = 0

Therefore, y = c1•e[tex]^{-t}[/tex]+c2•e[tex]^{-2t}[/tex]

2.) Use method of undetermined coefficients for RHS. Attempt Y = Ae[tex]^{2t}[/tex]+B. This means Y' = 2Ae[tex]^{2t}[/tex] and Y'' = 4Ae[tex]^{2t}[/tex]

Substituting it all in and simplifying, I get...

12Ae[tex]^{2t}[/tex]+2B = 3e[tex]^{2t}[/tex]+1

Comparing coefficients, I get the following...

12A = 3 => A = 1/4

2B = 1 => B = 1/2

So this would yield a particular solution of:

y = c1•e[tex]^{-2t}[/tex]+c2•e[tex]^{-t}[/tex]+(1/4)e[tex]^{2t}[/tex]+(1/2)

4.) Next, I take the derivative of the above equation, and then substitute the initial conditions. When I do this, I get a system of equations that yields the following constants. C1 = 3/4, C2 = -1/2.

5.) Then I just replace C1 and C2 in the equation for the particular solution.

Could somebody please tell me where I am going astray? Thank you VERY much!!!

I am unsure if I am going about the solution correctly.

1.) Find the characteristic equation.

r[tex]^{2}[/tex]+3r+2=0 [tex]\Rightarrow[/tex] (r + 1)(r + 2) = 0

Therefore, y = c1•e[tex]^{-t}[/tex]+c2•e[tex]^{-2t}[/tex]

2.) Use method of undetermined coefficients for RHS. Attempt Y = Ae[tex]^{2t}[/tex]+B. This means Y' = 2Ae[tex]^{2t}[/tex] and Y'' = 4Ae[tex]^{2t}[/tex]

Substituting it all in and simplifying, I get...

12Ae[tex]^{2t}[/tex]+2B = 3e[tex]^{2t}[/tex]+1

Comparing coefficients, I get the following...

12A = 3 => A = 1/4

2B = 1 => B = 1/2

So this would yield a particular solution of:

y = c1•e[tex]^{-2t}[/tex]+c2•e[tex]^{-t}[/tex]+(1/4)e[tex]^{2t}[/tex]+(1/2)

4.) Next, I take the derivative of the above equation, and then substitute the initial conditions. When I do this, I get a system of equations that yields the following constants. C1 = 3/4, C2 = -1/2.

5.) Then I just replace C1 and C2 in the equation for the particular solution.

Could somebody please tell me where I am going astray? Thank you VERY much!!!

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