Solve the initial value problem.
-$y ^ { \prime } + y = 2 e ^ { x } ; y ( 0 ) = 22$
A) $y = e ^ { x } + 21 e ^ { - x }$
B) $y = 2 e ^ { x } + 19 e ^ { - x }$
C) $y = 22 \mathrm { e } ^ { \mathrm { x } }$
D) $y = 4 e ^ { 2 } + 20 e ^ { - x }$

Question

Quizplus · Accepted Answer

The given differential equation is a first-order linear differential equation. To solve it, we can use the integrating factor method. The integrating factor, $ \mu(x) $, is $ e^{\int 1 dx} = e^x $. Multiplying the entire equation by $ e^x $ gives $ e^x y' + e^x y = 2e^{2x} $, which simplifies to $ \frac{d}{dx}(e^x y) = 2e^{2x} $. Integrating both sides with respect to $ x $ gives $ e^x y = e^{2x} + C $, where $ C $ is the constant of integration. Solving for $ y $ gives $ y = e^x + Ce^{-x} $. Using the initial condition $ y(0) = 22 $, we find $ C = 21 $, so the solution is $ y = e^x + 21e^{-x} $.

Solve the Initial Value Problem y′+y=2ex;y(0)=22y ^ { \prime } + y = 2 e ^ { x } ; y ( 0 ) = 22y′+y=2ex;y(0)=22

Solve the Initial Value Problem $y ^ { \prime } + y = 2 e ^ { x } ; y ( 0 ) = 22$