I only know how to do this with regular equations. How do I do this with a differential equation involved?
Any help would be greatly appreciated!
Thanks in advance!
You have not understood the definition.
The first term requires y(0), whci is zero. It is therefore not a non-zero term.
The second term is related to 8*sin(0) + 7*e^(0) = 7 -- This should lead you to the first non-zero term.
The third term whould be related to y".
You need values of the function and its derivatives. When they give you a derivative, don't be afraid of it.
Treating y as a constant, I get:
y'' = 7e^x = 7e^x
Yielding y(x) = 7x + 7x^2/2! + 7 x^3/3! but that is incorrect. I am thinking I shouldn't be treating y as constant but I don't know what else to do.
Let's write again the DE...
$\displaystyle y^{'}= 8\ \sin y + 7\ e^{x},\ y(0)=0$ (1)
Under the assumption that the solution is analytic in x=0, we can write...
$\displaystyle y(x)= \sum_{n=0}^{\infty} \frac{y^{(n)}(0)}{n!}\ x^{n}= \sum_{n=0}^{\infty} a_{n}\ x^{n}$ (2)
The derivatives of y in x=0 are computed as follows...
$\displaystyle y(0)=0 \implies a_{0}=0$
... and from (1)...
$\displaystyle y^{'}(0)= 7 \implies a_{1}=7$
Now we compute $\displaystyle y^{''}$ from (1)...
$\displaystyle y^{''}= 8\ \cos y\ y^{'} + 7\ e^{x}$ (3)
... and from (3)...
$\displaystyle y^{''}(0)= 8 \cdot 7 +7 = 63 \implies a_{2}= \frac{63}{2}$
Now You can proceed to compute succesive terms... if necessary...