Solve the differential equation y\" - 3y\' = 8e^2t using the Method of undetermi

Question

Solve the differential equation y" - 3y' = 8e^2t using the Method of undetermined coefficients. a. y (t) = c_1 + c_2 e^3t + 4e^2t b. y (t) = c_1 + c_2 e^-3t - 4e^2t c. y (t) = c_1 + c_2 e^3t - 2e^2t d. y (t) = c_1 + c_2 e^3t + 2e^2t e. y (t) = c_1 + c_2 e^3t - 4e^2t f. None of These Given the Initial-Value Problem y" + 5y' = 12e^t, subject to y (0) = 1 y' (0) = -13 where y_h = c_1 + c_2 e^-5t and y_p = 2e^t Find y (0.5), use e = 2.71828 a. -0.256 b. -0.456 c. -0.656 d. -1.256 e. -0.856 f. -1.056 Given y" - y' - 6y = e^3t, where y_h = c_1 e^-2t + c_2 e^3t. Find the correct expression that could be used to solve for y_p using the Method of Undetermined Coefficient. a. y_p = Ae^-2t b. y_p = Ae^3t c. y_p = At^2 e^3t d. y_p = Ate^3t e. y_p = Ate^f. None of These Given y" - 3y' = 6t, where y_h = c_1 + c_2 e^3t. Find the correct expression that could be used to solve for y_p using the Method of Undetermined Coefficients. a. y_p = At^2 b. y_p = At^2 + Bt^2 c. y_p = At^2 + Bt d. y_p = At e. y_p = At + B f. y_p = At + Bt

Explanation / Answer

From the given question,

1.y'' - 3y' =8e2t

complementary function

y'' - 3y' =0

The characteristic equation for this differential equation and its roots are

r2-3r=0

r(r-3)=0

r=0 and r=3

complementary solution is yc(t)=c1e0t+c2e3t

yc(t)=c1+c2e3t

To find particular integral

yp(t)=Ae2t ; y'=2Ae2t ; y''=4Ae2t

y'' - 3y' =8e2t

4Ae2t- 3 (2Ae2t )=8e2t

4A-6A=8

A=-4

yp(t)=-4e2t

y(t)=yc(t) + yp(t)

=c1+c2e3t-4e2t

correct option is (b)

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