Question

Solve the given integral equation or integro-differential equation for y(t). y'(t) +125 fit – vy(v) dv=5t, y(0) = 0 9 - 2010- y(t) =

Answer 1

Here (*) is convolution sign between two function

I Have done this by using laplace transform !

Thanks !

sowe i- Intro- differential Equation is + given gi(t) Tt) +125 St. 6 (tv) y (v) du : -40)=0 & Initial Conditions are as Intro we Com solve this differential Equation using Laplace Laplace transform Some Important Formulas, on T f & g of It-T) g Lt) Lt) dt @ Iconvolution operator) LA fag here fog can be function of any variable, So alow ue com say that. t t = v) , f /v) dv t. gt) CS Scannect with CamScanne= t-V

So, by equation St. y (4) +12rit ry en I Convolution operator Convolution of it and get * gt) It g (t) S z gv dv t * St. y't) +15 ( t * tt ) Apply haplace transform both side L&g10] +125 L t z Ly (t)} = 5 23 t } =&L {y} - y lo) + 125 x 1 , LEYLt)} 2 12 t } = As, L{ t } = 7 anti 1 So, 23 t Scanned with CamScanner CS हाना 82

but yo) = 0 I given) & Ly} -0 + 125 L {y} x 82 {y} 8 + 125 5 옻 82 82 = [ {y 83+ 125 sa 82 2 82 83 125 83 +125 Now Apply Inverse Laplece both side, 83 +125 5 83+ 125 As we have to find partias frection of 5 834125 5 83453 83 +125 18 +5 (82-58t as 2) CS Scanned-with CamScanner

A sts Bitc 82-58+25 18+5) (8²-55+25) 2) 5 = A (8²-58+25) (BSL) (sts) 8²/ A+B) + (-5A+5 B+c) s t 254150 A+B=0 -5A+5B4 c=0 25A+SL-5 2 A = 1 / 1 B = -1 7 3. - 1 st 2/ ists 15 B25 8-58+25 + yet B+125 d-8/15+23 15(3+5 (82-5512) Estiu L'his)}tt 82_Satos 5t 1 1. [18-9) is il- 15 15/2] + 8- 5/) 2 + 15 858+25 4 ICS Scanned with CamScanner 335541922-2003

탄 he t 너 1185) 54 2_ 18-5) 니 st 내 =) least 15 훈 업 15 + 1) + G Hd 2 by using shifting properly El { F677 = entf(4) Vt 1 5t 요 녁 이 7 I CS-5 Cam

Now using : Bos at E 32792 } Sinat 824925 a e Cos/53 t) + sylt - Le J 15 2 5%t é 53 y lt I est le 15 15 Los 1) 57. e 5 5 Site 2 Final Answer -st le ft) he JLE te Bio $13+ los 15J3t 553 Answer CS Scanned with CamScanner