Question

In Exercises 1–22, solve the differential equation.2y' - y = xe^(x/2)

Accepted Answer

Rewrite the given differential equation in standard linear form. Start with the equation: $$2y' - y = xe$$^{x/2}$$. Divide both sides by 2 to isolate y$$'$$: y$$' - \frac{1}{2}y = \frac{x}{2} $$e^{x/2}. $$Identify the integrating factor (IF) for the linear differential equation $$y' + P(x)y = Q(x)$$, where $$P(x) = -\frac{1}{2}. $$The integrating factor is given by $$\mu(x) = e$$^{\int P(x) \, dx}$$. Calculate $$\mu(x) = $$e^{\int -\frac{1}$${2} \, dx} = $$e^{-x/2}. $$Multiply both sides of the differential equation by the integrating factor $$e^{-x/2} to $$make the left side an exact derivative: $$e^{-x/2} y$$' - \frac{1}{2} $$e^{-x/2} y$$ = \frac{x}{2} $$e^{x/2}$$ $$e^{-x/2}. $$Simplify the right side to $$\frac{x}{2}. $$Recognize that the left side is the derivative of the product $$y e$$^{-x/2}$$, so write $$\frac{d}{dx} \left( y $$e^{-x/2}$$ \right) = \frac{x}{2}$$. Integrate both sides with respect to x$: $$\int \frac{d}{dx} \left( y $$e^{-x/2}$$ \right) dx = \int \frac{x}{2} \, dx$$. After integration, solve for y$ by multiplying both sides by e$$^{x/2}$$. This will give the general solution to the differential equation in terms of x$ and the constant of integration.

In Exercises 1–22, solve the differential equation.
2y' - y = xe^(x/2)

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주요 개념

First-Order Linear Differential Equations

Integrating Factor Method

Solving Nonhomogeneous Differential Equations