Question

Find y as a function of x if

y′′′-10y′′+21y′=24eˣ,
y(0)=18, y′(0)=21, y′′(0)=13
y(x)=________________

Answer 1

To solve for y as a function of x, we solve the characteristic equation from the homogeneous differential equation, propose a particular solution for the inhomogeneous equation, and apply the given initial conditions to determine the full specific solution.

Step-by-step explanation:

To find y as a function of x for the differential equation y''' - 10y'' + 21y' = 24ex, we need to solve the homogeneous equation first, and then find a particular solution for the inhomogeneous equation. The homogeneous equation is y''' - 10y'' + 21y' = 0.

First, solve the characteristic equation: r3 - 10r2 + 21r = 0. Factoring, we get (r - 3)(r - 7)r = 0. This gives us r = 0, 3, 7. Therefore, the general solution to the homogeneous equation is yh(x) = C1 + C2e3x + C3e7x.

Next, we find the particular solution to the inhomogeneous equation by proposing yp(x) = Aex and substituting it into the original differential equation to solve for A. After finding A, the particular solution yp(x) is added to the homogeneous solution yh(x).

Lastly, use the initial conditions y(0) = 18, y'(0) = 21, and y''(0) = 13 to solve for the constants C1, C2, C3. These values will give us the final specific solution y(x).