Reduction of Order. If a nontrivial solution f(x) is known

QUESTION:

Reduction of Order. If a nontrivial solution f(x) is known for the homogeneous equation the substitution y(x) = v(x)f(x) can be used to reduce the order of the equation, as was shown in Section 4.7 for second-order equations. By completing the following steps, demonstrate the method for the third-order equation given that f(x) = exis a solution.(a) Set y(x) = v(x)ex and compute y’,y’’ ,and y’’’.(b) Substitute your expressions from (a) into (35) to obtain a second-order equation in w: = v’(c) Solve the second-order equation in part (b) for w and integrate to find y. Determine two linearly independent choices for v, say, v1 and v2.(d) By part (c), the functions y1(x) = v1(x)ex and y2(x) = v2(x)ex are two solutions to (35). Verify that the three solutions ex,y1(x), and y2(x) are linearly independent on