Question

A third-order homogeneous lhear equation and three linearly independent solutions ate given below. Find a particular solution satisfying the given initial conditions y⁽³⁾+2y′′−y′−2y=0;y(0)=1,y′(0)=16,y′′(0)=0y1=ex,y2=e−x,y3=e−2x

Answer 1

To find a particular solution satisfying the given initial conditions for the third-order homogeneous linear equation, we can use the method of variation of parameters. This involves assuming a particular solution in a specific form, substituting it into the equation, solving for the parameters, and integrating to find the final solution.

Step-by-step explanation:

To find a particular solution satisfying the given initial conditions, we can use the method of variation of parameters. Let's assume the particular solution in the form yp(x) = u1(x)y1(x) + u2(x)y2(x) + u3(x)y3(x), where u1(x), u2(x), and u3(x) are functions to be determined.

1. Differentiate yp(x) three times to find its third derivative.
2. Substitute yp(x), y'p(x), y''p(x) into the given differential equation.
3. Set the resulting equation equal to zero and solve for u1'(x), u2'(x), and u3'(x).
4. Integrate the functions u1'(x), u2'(x), and u3'(x) to find u1(x), u2(x), and u3(x).
5. Return the particular solution yp(x) = u1(x)y1(x) + u2(x)y2(x) + u3(x)y3(x) as the final answer.