Question

#3 please write solutions neatly

3. (17 points) Find the general solution of the linear differential equation y" + 5y' + 4y = (3x - 8)e* using the method of undetermined coefficients.

Answer 1

C1 and C2 are arbitrary constants.

Let - einx. 3 Now the 2" +5g' +4y= (3x-8) ex i corresponding Homogeneous eanation of cis is 7" +50'+4)=0-> (ii) be a solution of lii) then from Gig we (ii) have m² tomt 4 = 0 .: (m+G) (m+1) = 0 in- - 1 - 4

In (x = cle-4x+ ca é x. is solution of (ii) Now since ms-i is root of (iii) the particular solution of (i will look like X Jp(x) = ( AX+B) x x e- (A x2 + Bx) ex then ap(x) = (2 Ax+B) e-x - (A x² + Bx) e- x 2A e-* - 22 Ax+B) e-X + Ax2+BXVe-X ap(x) = have from (i) we exf27 2A - 2( 24x+B+C AX2+BX) + 5(2 Ax+B) -514x2+BX) + 4(A x2+BX) (3x-8) e-x

*2(A - 5A +4A) + x(-4 A + B +10A-5 B+ 4B) + (24-2B +5B)} = (3x-8) 2X( 6A) + (24+3B) = 32-8 and 2A + 3 B= - 8 6 A= 3 3 B= – 8- 1 A = 2 .: B = -3 ap(x)= x x - 3) e-x. general solution of (is is given by f(x)= In (xit ap (2) JIR) - eleux text x(2-3) e-x x