Question

6. Solve the given initial value problem, with y0 = 2 and y'0) = 5: y" - 6y' + 5y = 20t +1

Answer 1

We have y" - by' + 5y = 20t + 1 The comes bonding auxiliary equation e 8²-68+5=0 & 2-58-&75 = 0 rls-5) -61-5)=0 (2-1262-5) = 0 and l = 5 » Hence the complementary scartina.ee Yo Lt): cieftçest Now, suppose that the particular solution is Yo (t) = At+B where A and B are to be determined y it A Y yct) 0 Plugging these values in the given diff. egn 6A + 5 At + 58= zot t1 5 At+ (5 B-6A) Compesing coefficients of t ² Constant ferm & = gott/ 5 A = 20 => A = 4 - 5B=1464 5B-6A 2/ = 25 » Bas Hence the particles solution is Jp (t) = 4+ + 5

Hence the general solution is y(t) = y(t) + Yp (t) cettCq e5t +46+5 » f'(t) = Get + 56ge5t + 4 Given. y 601 = 2 C + C + = 2 a) + ₂ -3 And y یع (ع) از 크 (+ 5 Cg +4=5 3 C, +502 = 1 from and we get and Ci=-4 C2=1 Hence the general solution is Ylt): -4et test + 4t + 5 And