Solve the following 2nd order differential equation explicitly by expanding up to order x5,

y''(x) = (1 + x^2)*y(x)

pls show working y =f(x)

To solve the given second-order differential equation explicitly by expanding up to order x^5, we can use the power series method. This method assumes that the solution can be expressed as a power series in x. Let's proceed with the solution step by step.

Step 1: Write the power series representation of the solution
Assume that the solution can be expressed as a power series in x:

y(x) = ∑[n=0 to ∞] a_n * x^n

where a_n represents the coefficients to be determined.

Step 2: Determine the derivatives of y(x)
First, let's calculate the first and second derivatives of y(x):

y'(x) = ∑[n=0 to ∞] a_n * n * x^(n-1)
y''(x) = ∑[n=0 to ∞] a_n * n * (n-1) * x^(n-2)

Step 3: Substitute the derivatives into the differential equation
Now, substitute the derivatives into the given second-order differential equation:

∑[n=0 to ∞] a_n * n * (n-1) * x^(n-2) = (1 + x^2) * ∑[n=0 to ∞] a_n * x^n

Step 4: Simplify the equation
To simplify the equation, we need to rewrite it in terms of x^n. For that, let's change the indices of the first term on the left side:

∑[n=0 to ∞] a_n * n * (n-1) * x^(n-2) = (1 + x^2) * ∑[n=2 to ∞] a_(n-2) * x^(n-2)

Now, we can rewrite the index of the second series as k = n - 2:

∑[n=0 to ∞] a_n * n * (n-1) * x^(n-2) = (1 + x^2) * ∑[k=0 to ∞] a_k * x^k

Step 5: Equate the coefficients of the same powers of x
Next, equate the coefficient of each power of x on both sides of the equation. Starting with the terms with n = 0 and n = 1, we get:

a_0 * 0 * (-1) * x^(-2) + a_1 * 1 * 0 * x^(-1) + ∑[n=2 to ∞] a_n * n * (n-1) * x^(n-2) = a_0 + a_1 * x + ∑[k=0 to ∞] a_k * x^k

Step 6: Solve for the coefficients
To determine the coefficients a_n, we can compare the coefficients of the same powers of x. Equating the coefficients, we get:

a_0 * 0 * (-1) = a_0 --> a_0 = 0
a_1 * 1 * 0 = a_1 --> a_1 = 0
n * (n-1) * a_n = a_(n-2) + a_k --> for n ≥ 2, where k + (n-2) = n

Using the recursive relationship, we can find the coefficients by calculating a_n in terms of previous coefficients a_(n-2) and a_k.

Step 7: Finding the values of a_n up to order x^5
Let's find the first few coefficients using the recursive relationship:

For n = 2:
2 * (2-1) * a_2 = a_0 + a_0 = 0 + 0 = 0
a_2 = 0

For n = 3:
3 * (3-1) * a_3 = a_1 + a_0 = 0 + 0 = 0
a_3 = 0

Continuing this pattern, we find that a_n = 0 for all odd values of n.

For n = 4:
4 * (4-1) * a_4 = a_2 + a_0
4 * 3 * a_4 = 0 + 0
a_4 = 0

For n = 5:
5 * (5-1) * a_5 = a_3 + a_1
5 * 4 * a_5 = 0 + 0
a_5 = 0

Therefore, for all even values of n greater than 2, the coefficients a_n are zero.

Step 8: Write y(x) in terms of the non-zero coefficients
The solution y(x) can now be written as:

y(x) = a_0 + a_1 * x + a_2 * x^2 + a_3 * x^3 + a_4 * x^4 + a_5 * x^5

Substituting the coefficients we found in Step 7, we have:

y(x) = 0 + 0 * x + 0 * x^2 + 0 * x^3 + 0 * x^4 + 0 * x^5

Simplifying this expression gives:

y(x) = 0

Therefore, the solution to the given second-order differential equation up to order x^5 is y(x) = 0.