Final answer:
To solve the Euler Initial Value Problem: 4x^2y''+8xy'+17y=0,y(1)=2;y'(1)=-3, use the power series method to find the recurrence relation for the coefficients a_n. Solve the recurrence relation to obtain the particular solution. Use the initial conditions to determine the values of the coefficients and find the complete solution.
Step-by-step explanation:
To solve the Euler Initial Value Problem: 4x^2y''+8xy'+17y=0,y(1)=2;y'(1)=-3, we can use the power series method.
We first assume that y has a power series representation given by y(x) = ∑(n=0 to ∞) a_nx^n. Substituting this into the differential equation and its derivatives, we can find the recurrence relation for the coefficients a_n.
By solving the recurrence relation, we can find a particular solution for the differential equation. Then, using the initial conditions y(1)=2 and y'(1)=-3, we can determine the values of the coefficients a_n and find the complete solution to the initial value problem.