Question

Using the fact that x 1 (t)=e t

is solution of the second order linear homogeneous DE (7+3t)x′′−3x ′−(4+3t)x=0 find a second linearly independent solution x 2
​ (t) using the method of reduction of order (Do NOT enter x 2 (t) as part of your answer) and then find the unique solution of the above DE satisfying the initial conditionsx(0)=−13,x (0)=15 Enter your answer as a symbolic function of t, as in these Do not include ' x(t)=
′in your answer. examples

Answer 1

To find a second solution to a differential equation using reduction of order, we start with a known solution and assume a new solution shaped by an unknown function multiplied with the known solution. After deriving a new differential equation for the unknown function and solving it, we can ascertain the second independent solution. Then, we apply initial conditions to find the constants for the general solution that satisfy these conditions, leading to the unique solution x(t).

Step-by-step explanation:

To find a second linearly independent solution to the second order linear homogeneous differential equation (7+3t)x′′–3x′–(4+3t)x=0, we use the method of reduction of order, starting with the known solution x1(t) = et. We assume a solution in the form x2(t) = v(t)et, with v(t) as an unknown function to be determined. Substituting this into the differential equation and utilizing the known solution's derivatives, we simplify to find a first-order differential equation for v(t), which can be solved to find v(t) and hence x2(t).

The unique solution that satisfies the initial conditions x(0) = -13 and x′(0) = 15 can be found by combining the solutions x1(t) and x2(t) as x(t) = C1x1(t) + C2x2(t), where C1 and C2 are constants determined by applying the initial conditions. After calculating these constants, we insert them back into the equation to obtain the unique solution x(t).