Question

Solve the differential equation: (10 points for each) a) x¨+5x˙+6x=0 x(0)=0.3,x˙(0)=0.02 b) x¨+16x˙+100x=5t x(0)=0,x˙(0)=0

Answer 1

To solve the given differential equations, one must find the roots of the characteristic quadratic equation for the homogeneous part, and then use initial conditions to find the specific constants. For the nonhomogeneous equation, a particular solution must be found and added to the general solution.

Step-by-step explanation:

To solve the given differential equations, we will use methods suitable for second-order linear differential equations with constant coefficients. For the first equation, which is homogeneous:

1. Solve the characteristic equation, which is quadratic in nature.
2. Find the roots of the characteristic equation to determine the general solution.
3. Apply the initial conditions to find the particular solution.

For the second nonhomogeneous equation:

1. Solve the associated homogeneous equation by finding the roots of the characteristic equation.
2. Apply an appropriate method, such as the method of undetermined coefficients, to find a particular solution for the inhomogeneous part.
3. Combine the general solution of the homogeneous part with the particular solution and apply initial conditions.