Nonlinear diff equation and power series

[Helpeme]

Homework Statement

i need to solve this diff equation.

y' = x² + y²

y = 1 when x = 0

i can assume that the answer is a power series on the form Ʃa_nxⁿ

andi only need the 4 first non zero terms of the power-series answer

Homework Equations

Ʃa_nxⁿ

The Attempt at a Solution

y = Ʃa_nxⁿ
y' = Ʃna_nx^(n-1)

Ʃn*a_nx^(n-1) - (Ʃa_nxⁿ)² = x²

i guess since the answer is only 4 first non zero term. i can write out the summation to 4 terms.

if i do that I am stuck with a bunch of a₁ a₂ a₃ ... and i can't mange to find the value of those coefficients. and i don't really know if that is the right way of doing it at all.

 

[mighty2000]

your approach to solving this differential equation would be to use mathematical methods and techniques to find a solution. Here is one possible approach:

1. Rewrite the equation as y' - y2 = x2.

2. Use the method of undetermined coefficients to guess a solution in the form of a power series, y = Ʃanxn.

3. Substitute this power series into the equation and equate coefficients of the same powers of x on both sides. This will give you a system of equations for the coefficients an.

4. Use the initial condition y(0) = 1 to solve for the first coefficient a0.

5. Once you have the first coefficient, substitute it back into the system of equations and solve for the remaining coefficients. You may need to use techniques such as substitution or elimination to solve the system.

6. Once you have all the coefficients, plug them into the power series solution and you will have your final answer.

Remember to only include the first four non-zero terms in your final solution, as specified in the problem. You can also use a computer algebra system or software to help you with the calculations.

Alternatively, you could also use other methods such as the method of Frobenius or the method of variation of parameters to solve this differential equation. It is always important to carefully check your work and make sure your solution satisfies the original equation and the initial condition.