Question

In Exercises \(11-16,\) solve the initial value problem. Confirm your answer by checking that it conforms to the slope field of the differential equation. $$\frac{d y}{d x}=2 x e^{-x}\( and \)y=3\( when \)x=0$$

Short answer

The solution to the initial value problem is \(y(x) = -2xe^{-x} + 2e^{-x} + 3\). This solution has been verified and conforms to the slope field of the differential equation.

Step-by-step solution

Integration

The first step is to perform integration to the differential equation \(\frac{dy}{dx} = 2xe^{-x}\). Use the formula for integration by parts, which states that \(\int u dv = u v - \int v du\), taking \(u = x \) and \(dv = 2e^{-x} dx\). Calculate \(dv\) and \(u\) that lead to \(du = dx\) and \(v = -2e^{-x}\). Finally, plug in these values into the integration by parts formula.

Simplify and solve for constant

The integration by parts formula gives \(y(x) = -2xe^{-x} + 2e^{-x} + C\), where \(C\) is the constant of integration. To solve for the constant, use the initial condition given as \(y=3\) at \(x=0\). Simply substitute these values into the equation to get \(C\).

Final Solution

Substitute \(C\) value back into the equation to obtain the final solution. This yields the final solution to the differential equation including the integration constant.

Verification

Verify the final answer by confirming that it aligns with the slope field of the differential equation. This means substituting the values of \(x\) into \(\frac{dy}{dx} = 2xe^{-x}\) and then comparing the results with the derivatives from the final solution of the differential equation.