Solve the equation using an integrating factor.
-$y ^ { \prime } - 2 y = - 2 t$ , t 0
A) y = t + 1/2 + C $e ^ { 2 t }$
B) y = t - C $e ^ { 2 t }$
C) y = Ct + $e ^ { 2 t }$
D) y = 1/2 + C $e ^ { - 2 t }$

Question

Solve the equation using an integrating factor. 
-$y ^ { \prime } - 2 y = - 2 t$ , t > 0
A) y = t + 1/2 + C $e ^ { 2 t }$
B) y = t - C $e ^ { 2 t }$
C) y = Ct + $e ^ { 2 t }$
D) y = 1/2 + C $e ^ { - 2 t }$

Quizplus · Accepted Answer

The integrating factor is $e^{-2t}$, multiplying both sides by this gives $(ye^{-2t})' = -2te^{-2t}$. Integrating both sides with respect to $t$ yields $ye^{-2t} = \int -2te^{-2t} dt$, which after integration by parts gives $ye^{-2t} = t/2 + C$. Multiplying both sides by $e^{2t}$ gives $y = t + 1/2 + Ce^{2t}$.

Solve the Equation Using an Integrating Factor y′−2y=−2ty ^ { \prime } - 2 y = - 2 ty′−2y=−2t

Solve the Equation Using an Integrating Factor $y ^ { \prime } - 2 y = - 2 t$