Let

Differentiating twice gives


When x = 0, we observe that y(0) = a₀ and y'(0) = a₁ can act as initial conditions.
Substitute these into the given differential equation:


Then the coefficients in the power series solution are governed by the recurrence relation,

Since the n-th coefficient depends on the (n - 2)-th coefficient, we split n into two cases.
• If n is even, then n = 2k for some integer k ≥ 0. Then




It should be easy enough to see that

• If n is odd, then n = 2k + 1 for some k ≥ 0. Then




so that

So, the overall series solution is


Number 4: line XZ; number 5: theory of congruent triangles.
Answer:

Step-by-step explanation:
First, you have to substitute both functions into the equation to get:

Then, you distribute the negative sign in front of g(x) to get:

And finally, you add/subtract the x's that have the same power to get:

The answer would be: t= -5/4