Question

+-15 points ZillDiffEQ9 6.R.010. My Notes Ask Your Teacher Use an appropriate infinite series method about x-o to find two solutions of the given differential equation. (Enter the first four nonzero terms for each linearly independent solution, if there are fewer than four nonzero terms then enter all terms. Some beginning terms have been provided for you.) y" - xy' - y = 0 Yp=x+ 3x3 + x + 105*7.

+-15 points ZillDiffEQ9 6.R.010. My Notes Ask Your Teacher Use an appropriate infinite series method about x-o to find two solutions of the given differential equation. (Enter the first four nonzero terms for each linearly independent solution, if there are fewer than four nonzero terms then enter all terms. Some beginning terms have been provided for you.) y" - xy' - y = 0 Yp=x+ 3x3 + x + 105*7.

Had an error with the app 6.9 is the problem I'm having

Answer 1

Given differenti al equation, By" - '- y=0) (1) Here x = 0 is an ordinary point of (1) since the coefficient of y" 0 at x = 0. Assume the solution of (1) to be y=Ž<«x* = 40 +Cjx+azx2 + axx? +arx* +a5x + 6px° +..+ axx* +... →(2) =y' = a1 +242x+3a3x2 +4Qqx + 5a3x* +626%* +...+nax*-*+... =y" = 2a, +2.3azx+3.4Qqx2 +4.5axx* +5.6067* + ... + (n − 1)na x*-* +... Substituting y, y'," in (1), we get {2a, +2. 3zx+3.40.72 +4.5a57 +5.600x* +...+(n-1)na x*-+... } - x{az +20x+3a3x2 +424x2 +5aşx* +60px" +...+ na x*-* +... } - {ag+ajx+azx2 +azx +aqx* + asx + @oxº +...+amx* +...} = 0. equating to zero the coefficients of the various powers of x, 24,-60 = 0 ie. az = (co-efficient of x° = 0) 2.30-&-4 = 0 ie. as = 2 (co-efficient of x'=0) In general, (x+1)(x + 2)a, 2-Ba, -44 =0(co-efficient of * = 0) =(n+1)(n+2)x+2 = (x+1),, is the recuurence relation. Be (n+1) +2 (n+1)(n+2)*** ag, n = 0,1,2,3,4,5,6.... →(3) now an ang mga tao (puting n=2 in (m) = = 4 za 24 4.5 2.3 2.3.4.52 May (putting n= 3 in (3)) 5 1.3 1.3.5 5.6 2.3.4"234564, (putting n= 4 in (3) 6 2.4 2.4.6 6.7 2.3.45412345674, (putting n=4 in (3) 27 6.7 now the general solution of (1) is y(x) = a, +a 4 2.4 S. 1.3.5 2.4.6 ,*' +23457 +234.5.69**+ 23.4.5.67€* = y(x)=co(1+*+ **23356*+... ) 2 x+ 2.4.6 2.3 2. 2.4 X + X + 2.3.4.5"2.3.4.5.6.7 1 4. 1 6. -x + x* + - +..... 2 2.4 2.4.6 1 = y(x) = do + +a l 11 1 z x+-x 3 1 + - 3 .5 +---x' +...., 3.5.7"