Let's walk through the step-by-step solution and **explain each step** for the given problem: --- ### **Step 1: Write the Differential Equation** Given: \[ x^2 y' + y = 0 \] Assume a power series solution of the form: \[ y(x) = \sum_{n=0}^\infty c_n x^n \] --- ### **Step 2: Compute the Derivative** The derivative is: \[ y'(x) = \sum_{n=1}^\infty n c_n x^{n-1} \] --- ### **Step 3: Substitute into the Differential Equation** Multiply \( y'(x) \) by \( x^2 \): \[ x^2 y'(x) = \sum_{n=1}^\infty n c_n x^{n+1-1} x^2 = \sum_{n=1}^\infty n c_n x^{n+1} \] But to match powers, shift the index: Let \( m = n+1 \Rightarrow n = m-1 \): \[ x^2 y'(x) = \sum_{m=2}^\infty (m-1) c_{m-1} x^{m} \] Now write \( y(x) \) in powers of \( x \): \[ y(x) = \sum_{n=0}^\infty c_n x^n \] --- ### **Step 4: Combine Terms** Combine: \[ x^2 y'(x) + y(x) = \sum_{n=2}^\infty (n-1) c_{n-1} x^n + c_0 + c_1 x + \sum_{n=2}^\infty c_n x^n \] \[ = c_0 + c_1 x + \sum_{n=2}^\infty \left[(n-1)c_{n-1} + c_n \right] x^n \] --- ### **Step 5: Set Coefficients to Zero** For this sum to be zero for all \( x \): - \( c_0 = 0 \)                                   [constant term] - \( c_1 = 0 \)                                   [coefficient of \( x \)] - For \( n \geq 2 \):                               [coefficient of \( x^n \)] \[ (n-1)c_{n-1} + c_n = 0 \implies c_n = -(n-1)c_{n-1} \] Since \( c_0 = 0 \), and by recursion, every subsequent coefficient is zero: - \( c_1 = 0 \) - \( c_2 = -(2-1)c_1 = -1 \cdot 0 = 0 \) - \( c_3 = -(3-1)c_2 = -2 \cdot 0 = 0 \) - and so on... --- ### **Step 6: Write the Solution** So all coefficients are zero: \[ y(x) = 0 \] --- ### **Final Answer** \[ \boxed{y(x) = 0} \] This is the only solution for the given power series form, as all coefficients must vanish. The image correctly shows this step-by-step process and verifies that the solution is \( y(x) = 0 \).

Certainly! Let's carefully go through each step to find the power series solution for the differential equation: \[ x^2 y' + y = 0 \] where the solution is assumed to be of the form: \[ y(x) = \sum_{n=0}^\infty c_n x^n \] --- ### **Step 1: Write the power series for \( y(x) \) and its derivative** Given: \[ y(x) = \sum_{n=0}^\infty c_n x^n \] The derivative: \[ y'(x) = \frac{d}{dx} \left( \sum_{n=0}^\infty c_n x^n \right) = \sum_{n=1}^\infty n c_n x^{n-1} \] --- ### **Step 2: Multiply \( y'(x) \) by \( x^2 \)** \[ x^2 y'(x) = x^2 \sum_{n=1}^\infty n c_n x^{n-1} = \sum_{n=1}^\infty n c_n x^{n-1 + 2} = \sum_{n=1}^\infty n c_n x^{n+1} \] Now, to express everything in terms of \( x^n \), shift the index: Let \( m = n + 1 \). Then \( n = m - 1 \), and when \( n = 1 \), \( m = 2 \): \[ x^2 y'(x) = \sum_{m=2}^\infty (m - 1) c_{m - 1} x^{m} \] --- ### **Step 3: Write the entire differential equation in terms of power series** The original equation: \[ x^2 y' + y = 0 \] Substitute the series: \[ \sum_{m=2}^\infty (m - 1) c_{m - 1} x^{m} + \sum_{n=0}^\infty c_n x^{n} = 0 \] Express the second sum to start from \( n=0 \): \[ \sum_{n=0}^\infty c_n x^{n} \] Now, for the first sum, rename \( m \) back to \( n \) for simplicity: \[ \sum_{n=2}^\infty (n - 1) c_{n - 1} x^{n} + \sum_{n=0}^\infty c_n x^{n} = 0 \] --- ### **Step 4: Combine sums, aligning powers of \( x \)** Express the entire sum as: \[ c_0 + c_1 x + \sum_{n=2}^\infty \left[ (n - 1) c_{n - 1} x^{n} + c_n x^{n} \right] = 0 \] Or: \[ c_0 + c_1 x + \sum_{n=2}^\infty \left[ (n - 1) c_{n - 1} + c_n \right] x^{n} = 0 \] --- ### **Step 5: Derive the recurrence relations** Since the power series equals zero for all \( x \), **each coefficient must be zero**: - Constant term (\( x^0 \)): \[ c_0 = 0 \] - Coefficient of \( x^1 \): \[ c_1 = 0 \] - For \( n \geq 2 \): \[ (n - 1) c_{n - 1} + c_n = 0 \] This is a recurrence relation to find \( c_n \) in terms of \( c_{n-1} \): \[ c_n = - (n - 1) c_{n - 1} \] --- ### **Step 6: Find the coefficients \( c_n \)** Given: \[ c_0 = 0, \quad c_1 = 0 \] Use the recurrence: \[ c_2 = - (2 - 1) c_1 = -1 \times 0 = 0 \] \[ c_3 = - (3 - 1) c_2 = -2 \times 0 = 0 \] \[ c_4 = - (4 - 1) c_3 = -3 \times 0 = 0 \] And so on. Inductively, all \( c_n \) are zero: \[ c_n = 0 \quad \text{for all } n \] --- ### **### **Final solution:** \[ \boxed{ y(x) = \sum_{n=0}^\infty c_n x^n = 0 } \] which indicates the **trivial solution**. --- ### **Summary of key points:** - The power series solution leads to all coefficients being zero. - The only solution is \( y(x) = 0 \). --- ### **Note:** In some cases, the differential equation may admit non-trivial solutions, but here, the initial conditions and the recurrence relations lead to the trivial solution only. --- If there's more context or specific boundary conditions, they could affect the solution, but based on the given series method, this is the complete step-by-step derivation.