Key Concepts and Formulas

• First-Order Linear Differential Equation: A differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$.
• Integrating Factor (IF): A function used to solve first-order linear differential equations, given by $\text{IF} = e^{\int P(x) \, dx}$.
• General Solution: The general solution to a first-order linear differential equation is given by $y \cdot (\text{IF}) = \int Q(x) \cdot (\text{IF}) \, dx + C$, where $C$ is the constant of integration.

Step-by-Step Solution

Step 1: Identify the Differential Equation and Standard Form

We are given the differential equation: $\frac{dy}{dx} - y = 1 + 4\sin x$ Our goal is to recognize this as a first-order linear differential equation and express it in standard form, which is $\frac{dy}{dx} + P(x)y = Q(x)$.

Comparing the given equation with the standard form, we identify: $P(x) = -1$ $Q(x) = 1 + 4\sin x$ This identification is crucial because $P(x)$ and $Q(x)$ are needed to calculate the integrating factor and subsequently solve the equation.

Step 2: Calculate the Integrating Factor (IF)

The integrating factor is calculated using the formula: $\text{IF} = e^{\int P(x) \, dx}$ Substituting $P(x) = -1$, we have: $\text{IF} = e^{\int -1 \, dx} = e^{-x}$ The integrating factor $e^{-x}$ will transform the left-hand side of the differential equation into the derivative of a product, making it directly integrable.

Step 3: Find the General Solution

The general solution of the first-order linear differential equation is given by: $y \cdot (\text{IF}) = \int Q(x) \cdot (\text{IF}) \, dx + C$ Substituting $\text{IF} = e^{-x}$ and $Q(x) = 1 + 4\sin x$, we get: $y e^{-x} = \int (1 + 4\sin x) e^{-x} \, dx + C$ Now, we need to evaluate the integral. We split it into two parts: $\int (1 + 4\sin x) e^{-x} \, dx = \int e^{-x} \, dx + 4 \int e^{-x} \sin x \, dx$

• Part A: $\int e^{-x} \, dx$ This is a standard integral: $\int e^{-x} \, dx = -e^{-x}$

• Part B: $\int e^{-x} \sin x \, dx$ We can use integration by parts twice, or the formula $\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2+b^2} (a \sin(bx) - b \cos(bx))$. Here, $a = -1$ and $b = 1$. $\int e^{-x} \sin x \, dx = \frac{e^{-x}}{(-1)^2 + 1^2} (-1 \cdot \sin x - 1 \cdot \cos x) = \frac{e^{-x}}{2} (-\sin x - \cos x)$ Thus, $4 \int e^{-x} \sin x \, dx = 4 \cdot \frac{e^{-x}}{2} (-\sin x - \cos x) = 2e^{-x} (-\sin x - \cos x)$

Combining the two parts: $\int (1 + 4\sin x) e^{-x} \, dx = -e^{-x} + 2e^{-x} (-\sin x - \cos x)$ Substituting this back into the general solution equation: $y e^{-x} = -e^{-x} - 2e^{-x} (\sin x + \cos x) + C$ Multiplying both sides by $e^x$: $y(x) = -1 - 2(\sin x + \cos x) + Ce^x$

Step 4: Apply the Initial Condition

We are given the initial condition $y(\pi) = 1$. We substitute $x = \pi$ and $y = 1$ into the general solution to find $C$: $1 = -1 - 2(\sin \pi + \cos \pi) + Ce^{\pi}$ Since $\sin \pi = 0$ and $\cos \pi = -1$: $1 = -1 - 2(0 - 1) + Ce^{\pi}$ $1 = -1 + 2 + Ce^{\pi}$ $1 = 1 + Ce^{\pi}$ $0 = Ce^{\pi}$ Since $e^{\pi} \neq 0$, we must have $C = 0$.

Substituting $C=0$ back into the general solution, we get the particular solution: $y(x) = -1 - 2(\sin x + \cos x)$

Step 5: Evaluate the Expression

We want to find the value of $y\left(\frac{\pi}{2}\right) + 10$. First, we find $y\left(\frac{\pi}{2}\right)$: $y\left(\frac{\pi}{2}\right) = -1 - 2\left(\sin \frac{\pi}{2} + \cos \frac{\pi}{2}\right)$ Since $\sin \frac{\pi}{2} = 1$ and $\cos \frac{\pi}{2} = 0$: $y\left(\frac{\pi}{2}\right) = -1 - 2(1 + 0) = -1 - 2 = -3$ Now, we compute $y\left(\frac{\pi}{2}\right) + 10$: $y\left(\frac{\pi}{2}\right) + 10 = -3 + 10 = 7$

Common Mistakes & Tips

• Remember to include the constant of integration when finding the general solution.
• Be careful with signs when calculating the integrating factor and evaluating integrals.
• Double-check the values of trigonometric functions at specific angles.

Summary

We solved the first-order linear differential equation using the integrating factor method. After finding the general solution, we used the initial condition to find the particular solution. Finally, we evaluated the expression $y\left(\frac{\pi}{2}\right) + 10$ using the particular solution, which resulted in 7.

The final answer is $\boxed{7}$.