Question

Solve the initial value problem:

(4 + x²) y'' + 2y = 0, y(0) = 0, y'(θ) = 6
If the solution is y = c₀ + c₁x + c₂x² + c₃x³ + c₄x⁴ + c₅x⁵.

Answer 1

To solve the initial value problem (4 + x²) y'' + 2y = 0 with conditions y(0) = 0 and y'(θ) = 6, one must differentiate the proposed polynomial solution, match coefficients after substitution into the differential equation, and apply the initial conditions to find the constants.

Step-by-step explanation:

The initial value problem given is (4 + x²) y'' + 2y = 0, with initial conditions y(0) = 0, and y'(θ) = 6. The student is looking for a solution in the form of a polynomial y = c₀ + c₁x + c₂x² + c₃x³ + c₄x⁴ + c₅x⁵. To solve this, we must differentiate the proposed solution to obtain expressions for y' and y'', then substitute these into the differential equation and match the coefficients to satisfy the equation. The initial conditions help us determine the constants c₀, c₁, etc. Properly determining the constants will ensure the initial conditions are met, leading to the specific solution for the given problem.

The process of solving this involves:

1. Finding the first and second derivatives of the proposed solution.
2. Substituting these derivatives into the initial value problem's differential equation.
3. Matching coefficients to find the values of constants c₀, c₁, etc. that satisfy both the differential equation and the initial conditions.

There seems to be some confusion with unrelated text in the question details, but the primary focus for solving this problem is applying knowledge on differential equations and initial value problems, a staple in the mathematics field, particularly at the college level.