Question

Solve (D^3+1)y=3+e^-x

Answer 1

To solve the equation (D^3+1)y=3+e^-x, we can proceed by isolating the terms involving the differential operator D on one side and solving it as a polynomial equation. The general solution to the equation is y=C1e^(1.26x)+C2e^(1.26xcos(1.26x))+C3e^(1.26xsin(1.26x)).

Step-by-step explanation:

To solve the equation (D^3+1)y = 3+e^-x, we can start by rewriting it as (D^3+1)y - e^-x = 3. This equation involves the differential operator D and the exponential term e^-x. To solve it, we can consider the operator D as a variable and solve for y using algebraic techniques. Let's proceed to solve this equation step by step.

1. Start by isolating the terms involving the differential operator D on one side:
2. (D^3+1)y = 3+e^-x - e^-x
3. (D^3+1)y = 3
4. Next, solve the equation as a polynomial equation, considering D^3 as a single variable:
5. D^3y + y = 3
6. This is a linear homogeneous ordinary differential equation. Let's substitute y = e^rx:
7. (r^3 + 1)e^rx = 3e^rx
8. Divide both sides by e^rx:
9. r^3 + 1 = 3
10. Solve this cubic equation for r:
11. r^3 = 2
12. r = ∛(2) or approximately 1.26
13. Thus, the general solution to the equation is y = C1e^(1.26x) + C2e^(1.26xcos(1.26x)) + C3e^(1.26xsin(1.26x)), where C1, C2, and C3 are constants.

This is the solution to the given equation (D^3+1)y = 3+e^-x.

Learn more about Solving differential equations with differential operator D and exponential term e^-x