Question

15. Use the method of undetermined coefficients to find a particular solution to the equation below (you must solve for all the constants!). Then use your particular solution to find a general solution to the equation (give an explicit final answer in the form “y = ..."). dy day · +37-10y = 30t2 dt2 dt

Answer 1

The given ven differential equation is dy dy dar + 3 30tr at - loy 2 = . The homogeneous part of ou. de ten + 3 do -1oy = auxiliary this in equation for The mrt om lo 7 mrt 5m-Rm-10 =0 7 mlm+5) -2(m+5) (m+5) m-2) = 0 m = -52. Hence, complementary function of ☺ where a q are content. Yc cest te eet by Now we find particular integral of o. method of tender undetermined co-efficient , Now let trial solution is our ар Ait²+ Bt + c dyp - 2 At + B at diye 2A

Now putting these values of the in we get, 2A + 3(2 A++B) - 10/4+48+ + 9) = 30+ 2A + GA & +38-10 A t - 10Bt - loc 30 ca (2A +38-106 ) +108) + + €104) * = 304" + we get, comporting like powers of 2 A43B-100 O -JOB (LOA) = 30 ZO ヲ A = -3, 2+(-3) + 3X0 JOC=0 6-10C. 0 AOC = 6 olo - C = 3 5 Yp = -3+ + 0.7 - yo = - 3t Malw Yp The general solution is y = cest + 2 0 2 + 3t 3 5 Where a,q are constants