Solve the initial value problem.
-$$\frac { d y } { d t } + 7 y = 3 ; y ( 0 ) = 1$$
A) $y = \frac { 3 } { 7 } e ^ { - 7 t } + \frac { 4 } { 7 }$
B) $y = \frac { 4 } { 7 } e ^ { 7 t } + \frac { 3 } { 7 }$
C) $y = \frac { 4 } { 7 } e ^ { - 7 t } + \frac { 3 } { 7 }$
D) $y = \frac { 3 } { 7 } e ^ { 7 t } + \frac { 4 } { 7 }$

Question

Quizplus · Accepted Answer

This is a first-order linear differential equation. The solution to such an equation can be found using the integrating factor method. The integrating factor, $e^{\int 7 dt} = e^{7t}$, is used to multiply both sides of the equation, allowing it to be written in the form $d(e^{7t}y)/dt = 3e^{7t}$. Integrating both sides with respect to $t$ and applying the initial condition $y(0) = 1$ leads to the solution $y = \frac{4}{7}e^{-7t} + \frac{3}{7}$, which matches option C.

Solve the Initial Value Problem dydt+7y=3;y(0)=1\frac { d y } { d t } + 7 y = 3 ; y ( 0 ) = 1dtdy​+7y=3;y(0)=1

Solve the Initial Value Problem $\frac { d y } { d t } + 7 y = 3 ; y ( 0 ) = 1$