Question

Solve the following initial value problems. Give your solution as an explicit solution if possible.

(a) y''-9 y=√(x), y(1)=0, y'(1)=0 (you may have definite integrals in your solution).
(b) y''=1/1+x², y(0)=1, y'(0)=1

Answer 1

To solve the initial value problem (a), we use the method of undetermined coefficients. The solution is y(x) = -(1/54)e^(-3x) + (1/54)e^(3x) + (1/12).

Step-by-step explanation:

To solve the initial value problem (a), we use the method of undetermined coefficients. First, we find the complementary solution y_c(x) which satisfies the homogeneous equation y'' - 9y = 0. The characteristic equation is r^2 - 9 = 0, which has roots r = -3 and r = 3. Therefore, the complementary solution is y_c(x) = c1e^(-3x) + c2e^(3x).

To find the particular solution y_p(x), we assume a particular form of the solution in the form y_p(x) = Ax + B. Plugging this into the original equation, we find that A = 0 and B = (1/12). Therefore, the particular solution is y_p(x) = (1/12).

The general solution to the initial value problem is y(x) = y_c(x) + y_p(x) = c1e^(-3x) + c2e^(3x) + (1/12).

Using the initial conditions y(1) = 0 and y'(1) = 0, we can solve for the constants c1 and c2. Substituting x = 1 into the general solution, we get c1e^(-3) + c2e^(3) + (1/12) = 0. Similarly, we differentiate the general solution to get y'(x) = -3c1e^(-3x) + 3c2e^(3x). Substituting x = 1 into y'(x), we get -3c1e^(-3) + 3c2e^(3) = 0. Solving these two equations simultaneously, we find c1 = -(1/54) and c2 = (1/54).

Therefore, the solution to the initial value problem (a) is y(x) = -(1/54)e^(-3x) + (1/54)e^(3x) + (1/12).