Question

Find the particular solution of the differential equation that satisfies the initial conditions. f ''(x) = 2 cos(x), f '(0) = 4, f(0) = −9

a) f(x) = -3sin(x) + 4cos(x) - 9
b) f(x) = -3sin(x) - 4cos(x) - 9
c) f(x) = 3sin(x) + 4cos(x) - 9
d) f(x) = 3sin(x) - 4cos(x) - 9

Answer 1

To find the particular solution of the differential equation, we can integrate both sides of the equation with respect to x and apply the initial conditions. The particular solution is found to be f(x) = -3sin(x) + 4cos(x) - 9.

Step-by-step explanation:

To solve the given differential equation and find the particular solution that satisfies the initial conditions, we can integrate both sides of the equation twice. Integrate the right side of the equation with respect to x:

• ∫ f''(x) dx = ∫ 2 cos(x) dx
• f'(x) = 2 sin(x) + C1

Integrate the right side of the equation again with respect to x:

• ∫ f'(x) dx = ∫ (2 sin(x) + C1) dx
• f(x) = -2 cos(x) + C1x + C2

Using the initial conditions f'(0) = 4 and f(0) = -9, we can substitute these values into the equation to find the particular solution.

After substituting the initial conditions and simplifying, we get the particular solution as f(x) = -3sin(x) + 4cos(x) - 9 (option a).