Question

7. Solve the second order differential equation (you can use any method ) y" + y = csc(x)

Answer 1

To Solve This Differential equation i have used Method of variation of parameters !

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Explanation :-

Solution : given → Here, differential équation is on (sco) y" ty Tsc (x) = Cosec (o). Here, y = day dar Sirp). can solve this differential Equation using Method of Variation of parameters We So. In the form. iven equation is Y" + plasy q (x) y r(a) In our DE (Differential Equation) Plat) = 0, ql2)=1, Vla) Cosecha. As, According to the definition of Linear ODE, + Pexes y + q lxgy ra also Linear ODE- yu our of Linear ODE is given by Yo (x) + 9,14) glas & Solution of Ho Scanned with Homogenous OSE Camśtahad :- perticular solution equation is general solution Y(a) CS

and Ip. Lets take to the equatim. that is = So, How to find year - for Yh (2) Linear Homogenou enous Equation corresponding Linear y" + plasy 962g = rlas So, tosresponding tomogenous Equation will be y" + plas y + q lsely zo So, BODE'S Homogenous part is y" ty Its basis of solutions can be obtained roots of charactersties equation 1+1 = 0 x= t. our So, Ipad roots a = ti y (x) = A Cos dx + B Sinda So, In A loset B sinx Cs Scanned with (A 2 B7 Age Constants. CamScanner

So, basis of solutions of Homogenous ODE Cosa y = yz = sina → To find Yplay da. Langrange Methood gives perticular solution up. yology, (a (2) tat 12 ) = 0 wilonskian of y o y y y - y y y y Yr W= ODE Now for our yuty Cosec (a]. We have from ye y Cost Sina Losa, y = sina Cos²x -|- Sin2) (losse) - Sina Cosx Hesse)! W = Cos²x & Sis 2x Scanned with CamScanner CS

So, y (x) = -y, f g ray dx + y y, rlos det but y = Cosa, y = Sind, W=1, rlas = cosees. - yola) = flos sy binia). Cosecx) dx + dar 1 1 upla) = -lossi (1) dx + Sina 1) dx + Sina tota da. [. (Sina). (Coseca) (Béna). I a losa). Coseca 1 sina tosa). Coseca) = Cosu). I -lota Besa) plas (-Cosx) . (2) + (Send) lag is a Plex s dx=a I lota du = log/senx) Jp (x) = -x los c) + Siste) In señal de ན་ So, general solution y(x) = {Ye (oc), typ 102 A losko+ B Sen [29] + & Cos(2) + Simp) in Sint i 1200 W463(x) + B8in (2) -xlosla + sin b) In sinal tamscanner

Answer is : A cos(x) + B sin(x) - (x)cos(x) + sin(x)ln|sin(x)| .