Question

Q1 (10 points) Consider the differential equation ty" _ y = 0. a) is this differential equation linear? What is its order? Is it homogeneous? b) Try a solution of the form y=x". Is this a solution for some r? If so, find all such r. c) Based on your answer to a) about linearity and b) about what y=x" are solutions, make an educated guess a the general solution looks like. Try that guess and check that it works. Brad and drop VIOHE

Answer 1

Sony Given differential equation, = 0. Q) xy" - y'=0 A differential equation is Called linear if clependent variable and. derivative involved occurs in the first degree only and () No products of dependent Variable and / or derivative occur every eve

livomerilicom property Since the differential equation equation Satisity both the above m. Therefore given different al equation is. linear differential equation. Order of differential equation is the highest derivalive in the equatin occur Idence order of differential equation is. 2. an dny either a is a only chitterential A linear differential equation of the form a, ani dnoty the tany=X dan dan- (2 Where apar .. an are Constants and x is Constant or a function of a Called neous linear equation. Multiply eno by 'x' We have x2 dy x dy О - ela? da Comparing with ③ we have n=2 :41,= -1 and a=0 x=0 therefore egn Represents a Homgenery differential Qualiom. Consequently ① is a homogeneous.

(6) 164 y = x is of the form of solution of differentiating with go respect to t dy da xxr-1 4 Again diffrerentiating we get dy r(0-1) xr-2 da? in o. =0 8-2t 0 o rx } =0 Putting values of 4 and 5 We have alr(0-1) x 8-2) rx8-1) or {x(x-1) 8th or { 8(0-1) x at {8(8-1) -8} xot Since x 80 therefore we have &&(8-1)-8}=0. r(8-1-1) rir-2) =0 or o 8=0 and r=2

Any Yes, for and r=2 8=0 y = x is the solution of equeation . qiuen differential. (0) Given equation xy" - y' or dy so a dy da² da and so q Put dy dq dzy da? are then we da have q = 0 or dre 2. e da da da q get. da q Integrating he f da re + Log C Log 2 Log q Replacing Log x log xc. acf or q by All have dy da x C dy dx

both side We have Integerating ( dy = cf a are or y = e x2 + 2 Augs tok: General solution of ① like ly cx? tc where 2 c and ' are being ar bitrary Constant y = x² y = x=1 and Clearly is the two linearly independent Solution of for c= 2,6=0 and C=0 C-1 in the above general solution, Therefore it works gespectively