Let's solve **Question 1** step step: --- **Step 1: Write the Differential Equation and Identify the Power Series Solution** Given: \[ (1 + x^2)y'' - 10x y' + 28y = \] We seek a power series solution near \( x = \) of the form: \[ y(x) = \sum_{n=}^{\infty} a_n x^n \] **Explanation:** We assume the solution can be written as a power series centered at \( x = \), which is standard for finding solutions near an ordinary point. --- ## **Step 2: Find the Minimum Radius of Convergence** The minimum radius of convergence is determined by the distance from \( x = \) to the nearest singularity of the equation. A singularity occurs where the coefficient of \( y'' \), i.e., \( 1 + x^2 = \). So, set \( 1 + x^2 = \implies x^2 = -1 \implies x = i \) or \( x = -i \). The distance from \( x = \) to \( x = i \) (or \( x = -i \)) is \( |i| = 1 \). **Explanation:** The singular points are at \( x = i \) and \( x = -i \). The radius of convergence is the distance from the center () to the nearest singularity in the complex plane. --- ## **Final Answer** **The minimum radius of convergence of the power series solutions is \( \boxed{1} \).**

## Question 1: Power Series Solutions ### Step 1: Write the Differential Equation and Identify the Power Series Solution Given the differential equation: \[ (1 + 2)y'' - 10zy' + 28y = 0 \] We will assume a power series solution around \( z = 0 \): \[ y(z) = \sum_{n=0}^{\infty} a_n z^n \] **Explanation:** This step lays the groundwork for finding a solution in terms of a power series centered at the ordinary point \( z = 0 \). The coefficients \( a_n \) will need to be determined by substituting the power series into the differential equation. --- ### Step 2: Find the Minimum Radius of Convergence Next, we identify the singular points of the differential equation. The term \( (1 + 2) \) indicates that the coefficient of \( y'' \) can become zero. We solve: \[ 1 + 2 = 0 \implies z = -\frac{1}{2} \] The singular point is at \( z = -\frac{1}{2} \). **Distance Calculation:** The distance from \( z = 0 \) to the nearest singularity \( z = -\frac{1}{2} \) is: \[ |- \frac{1}{2} - 0| = \frac{1}{2} \] **Explanation:** The radius of convergence is determined by the distance to the nearest singularity. Since \( z = -\frac{1}{2} \) is the closest point at which the differential equation ceases to be analytic, the radius of convergence is \( \frac{1}{2} \). --- ## Final Answer **The minimum radius of convergence of the power series solutions is \( \boxed{\frac{1}{2}} \).** --- ## Question 2: General Solutions ### Step 1: Write the Differential Equation Given: \[ y'' = 3y + 7y' - d(z - 2) \] Rearranging gives: \[ y'' - 7y' - 3y = -d(z - 2) \] **Explanation:** This standard form allows us to apply methods for solving linear differential equations. --- ### Step 2: Find the General Solution The corresponding homogeneous equation is: \[ y'' - 7y' - 3y = 0 \] To solve it, we find the characteristic equation: \[ r^2 - 7r - 3 = 0 \] Using the quadratic formula: \[ r = \frac{7 \pm \sqrt{49 + 12}}{2} = \frac{7 \pm \sqrt{61}}{2} \] Thus, the general solution to the homogeneous equation is: \[ y_h = C_1 e^{\frac{7 + \sqrt{61}}{2} z} + C_2 e^{\frac{7 - \sqrt{61}}{2} z} \] For the non-homogeneous part, we will use the method of undetermined coefficients or variation of parameters based on \( -d(z - 2) \). **Explanation:** We first tackle the homogeneous part, yielding two solutions based on the roots of the characteristic polynomial, and then apply a method to determine a particular solution. --- ## Final Answer **The general solution is:** \[ y(z) = C_1 e^{\frac{7 + \sqrt{61}}{2} z} + C_2 e^{\frac{7 - \sqrt{61}}{2} z} + y_p \] --- ## Question 3: Solve the Initial Value Problem ### Step 1: Write the Equation Given: \[ y^{(4)} + a^2 = 0 \] With initial conditions \( y(0) = 0, y'(0) = 1, y''(0) = 0, y'''(0) = 0 \). **Explanation:** This is a fourth-order linear homogeneous differential equation where we need to find a function \( y(z) \). --- ### Step 2: Solve the Differential Equation The characteristic equation is: \[ r^4 + a^2 = 0 \] Which gives roots: \[ r = \pm \frac{\sqrt{2}}{2} i \] The general solution is: \[ y(z) = A \cos(a z) + B \sin(a z) \] Applying initial conditions: - \( y(0) = 0 \) implies \( A = 0 \). - \( y'(0) = 1 \) implies \( B = \frac{1}{a} \). Thus, the solution becomes: \[ y(z) = \frac{1}{a} \sin(a z) \] **Explanation:** The roots indicate harmonic solutions. The initial conditions lead to specific coefficients, allowing us to construct the particular solution. --- ## Final Answer **The solution to the initial value problem is:** \[ y(z) = \frac{1}{a} \sin(a z) \] --- ## Question 4: Solve the Initial Value Problem ### Step 1: Write the Equation Given: \[ y'' + 8y' + 16y = f(t) \] With initial conditions \( y(0) = 1, y'(0) = 3 \), and \( f(t) \) defined piecewise. **Explanation:** This second-order linear differential equation requires a method to find a particular solution along with the homogeneous solution. --- ### Step 2: Solve for the Homogeneous and Particular Solutions The homogeneous part: \[ y_h'' + 8y_h' + 16y_h = 0 \] The characteristic equation: \[ r^2 + 8r + 16 = 0 \implies (r + 4)^2 = 0 \implies r = -4 \text{ (double root)} \] Thus, the homogeneous solution: \[ y_h(t) = (C_1 + C_2 t)e^{-4t} \] For \( f(t) \), we will find a particular solution based on the form of \( f(t) \). **Explanation:** The roots indicate repeated roots leading to the form of the homogeneous solution, and a particular solution will be crafted based on the forcing function. --- ## Final Answer **The complete solution will be of the form:** \[ y(t) = (C_1 + C_2 t)e^{-4t} + y_p(t) \] --- ## Question 5: Solve the Initial Value Problem ### Step 1: Write the Equation Given: \[ y'' + 9y = f(t) \] With initial conditions \( y(0) = 0, y'(0) = 1 \) and \( f(t) = \cos(t) \) for \( 0 < t < 1 \). **Explanation:** This is a second-order linear differential equation, and we need to find both the homogeneous and particular solutions. --- ### Step 2: Solve for the Homogeneous and Particular Solutions The homogeneous part: \[ y_h'' + 9y_h = 0 \] The characteristic equation: \[ r^2 + 9 = 0 \implies r = \pm 3i \] Thus, the homogeneous solution is: \[ y_h(t) = C_1 \cos(3t) + C_2 \sin(3t) \] For the non-homogeneous part, we try a particular solution: \[ y_p(t) = A \cos(t) + B \sin(t) \] Substituting \( y_p \) into the differential equation allows us to solve for \( A \) and \( B \). **Explanation:** The roots give rise to oscillatory homogeneous solutions, and we then find coefficients for the particular solution based on the form of \( f(t) \). --- ## Final Answer **The solution to the initial value problem will have the form:** \[ y(t) = C_1 \cos(3t) + C_2 \sin(3t) + y_p(t) \] With \( y_p(t) \) determined by substitution into the differential equation.