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


 
        
             
        
        
        
13 factored out is 13x - 39 add 39 to both sides it equals 13x=78 and 78 divided by 13 is 6 the anwser is x=6
        
             
        
        
        
Answer: The given question is solved :
Step-by-step explanation:
correct solution in imaje