Question

Differential equation question

please I need help with this question. Please show all work with clear hand writing

Find the solution of the second order differential equations: day + y = 0, y(TT/3) 0, y'(TT/3) = 4 dx2 a. = b. y" – 8y' + 16y = 0, y(0) = 1, y(1) = 0

Answer 1

u The given differential Equation is day dee? + y = 0 with initial condition y(1113) =0 f y'enle) = 4 So, quvilalęy equation of equation is m?+1=0 ie m²=-1 lie M=+ m-te So, it's gepleie complementeey function is Y(Q): 4p(06(ra) + (2 611) ; where a fce972 Babitasy lie Constant. yla) = cpcoolæ) + (2_010): which is required general solution to the equation

put cn = 2 in ez? O then 0+(2-0 becomes Ito (2) = 0 (15 - 26 so, finally put values of C1 = - 2 Jo ļ 62= 2 in y(a) then ya) = -259006(a) + 2610mm) which is roquired general solution to the given differential eq?
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so it'o general Solution is 412) - (9+(20) Jee Here the given differential equation is y" -8y +164 = 0 with initial condition 460)=1 , Yll)=0 So, quxilajey equation of equation ®ig m2_8m+16=0 le m? 264m) +02-0 ie (m-4)2 = 0 lie m= 4,4 ię 41a) = ((1+62 @)e"; (1, en-eebitavy Constant. which is requited general solution to the equation Da

but we know Y10) -1, 50, put a zo in ytre) then y(x) =(1+62.2)e for be comes Y(x=0) = ((1+(2.0) x 0 = (atole y=0) = a Ş: 410) =) lie C1 = 1 then we differentiate yin with respective to me then y'ın) = cietoa Hel8 (1+(218) Hoy + - ((1+461) ete+ Hoone HE yline) - 5016 the + 4 Corsetre we know y'18) -0,, so put net nya) then 412) becomes y'in =1) - 50 € 4X1 + 4 (alle 4x1 5964 + 4 czet 4!11) =D in but (1=1 then 5x1x8t it doet lp (264 - 504 4 then given you becomes Yia) = (1+(-3)2) e 12 which is required general selection to the given op * o

Note: In example (b) we use x as independent variable