Question

I need help with this question of Differential Equation.

Thanks

Use variation of parameters to solve the following system of ordinary differential equations. (dxlat = 2x - y dy dt = 3x - 2y + 4t

Answer 1

The given sistem of ordinary differential equations da at za y 0 dy at 3 n - rytat (2 from ② , we ger (D - 2) a +820 +(012) [ D= d af - 32 - 4 t we ger oparating (02) in (3) then subtracting (on) (0-2) + (0) + 3x - (0+2) y 20- 4t → (0%. 4) = - 4t (0?-1) a = - 4t Auxilary is by + 3a Then equation given mr 1 = 0 2) m = tl def = C, e te ēt where c. are arbitrary constanta using the method of variation of parameters, we ger vi et + og et 2 p. I es

Where U are given by 6, (-4t). ět dt از رام ) لما Not and 12= S (-At) et dr wlet, et NOTE Lex be two Y Y linearly independent saluation y" + Plt) y + 8(t) = f(t) then of Yol vyt U2 Y and v, f(t). Ja dt لا لا له .. j v2 (وه ره لا ) لنا .. dt. I gowy and wily, gr) yi YI Now weet, et ) et let) etcēt) -2 =>(4+). et dt =-2 / tēt dt = -2 [teet) - e'] 22 ēt [t+1

4, = 2 et [tri = ft4t). et dt 2 = 2 ste t et at = 2 (tet – et ) (t-1) V z 2 et e [t-1 = 2 et → 2.1 = 2 é '[t+iJe + ze [t-1] ēt (2 + + 2 ) +(2+-2) Eby®] u = 4 t a=2 e.F. + aprI c, ett tejet + 4 t

putting this Value of a no 2 we ger (0-2) (e, et eget + 46) 47-0 = y = 2(ci et tiez et + 4E) -(e, et èt +4) 2) ly= y = e, et + 34 et + 8t - 9