Question

Find the general solution of the given differential equation. (1 + 2?)yı - 2xy = 21(1 + r)

A. (1 + r)e^(2x)
B. (1 + r)e^(-2x)
C. (1 - r)e^(2x)
D. (1 - r)e^(-2x)

Answer 1

To find the general solution of the given differential equation, we first need to solve it using the method of power series. The general solution will have the form y(x) = c1(1 + r)e^(2x) + c2(1 - r)e^(-2x), where c1 and c2 are arbitrary constants. Hence the correct answer is option D

Step-by-step explanation:

To find the general solution of the given differential equation, we first need to solve it. The equation is (1 + 2r)y'' - 2xy = 21(1 + r). This is a second-order linear homogeneous differential equation with variable coefficients. One way to solve this equation is by using the method of power series.

We assume a power series solution of the form y(x) = ∑[n=0]^(∞)a_nx^n. Substituting this into the differential equation and equating coefficients of like powers of x, we can find a recurrence relation for the coefficients an.

By solving the recurrence relation, we can find the general solution of the differential equation. The general solution will have the form y(x) = c1(1 + r)e^(2x) + c2(1 - r)e^(-2x), where c1 and c2 are arbitrary constants.

Therefore, the correct answer is option D. (1 - r)e^(-2x).