Solutions 1. How many boundary conditions are needed to uniquely determine y, given d2 dx 2 y = - y + x +2 ? This is a second order D. E. and therefore we need 2 B.C.S. 2. Which of the following equations are linear? (3 0 " dx3y = y, gy () = - y2, d2 dzz y(z) = sin (y) For @ Set y = ay, +by 2 while y, and Y2 are the solutions of equations. 13 < L. H. S. ) = .> = Jas y 11 (ay,+by2) dx 3 For (2) 1. H.S.> = dg = q (ag +by2) = a de dx3 9 1 + b 9 5 by 3 y 2 = a. y, + by z Therefore egno is linear. = a dy , +baty2 = a. (-9,2) + b (-y2) But <R.H.S.> = - y' =- (ay, +by2) 2 = - azy, 2_bzy2 + 2aby , y z Therefore < L . H . S . > > < R . H. S. > Thus egne is NOT Linear.
12 For 3, < LHS > = y = 72 4 d 12 (ay,+by2) = a + y 1 + b = y 2 = a. sin(y, ) + b. sin ( 2). <R.H.S.>= Sin(y) = Sin (ay,+by z ) : < LHS> > <RHS> Thus egn 3 is NOT Linear!
3. Find the Taylor series of exp(x 2 ) around x 0 = 0. We will find the first 4 terms of the Taylor Series . f(x)= exp [x2] f(x) = 2x. exp[x2] f(x)= 2.exp[x2]+ 4x2. exp [x2] f"(x) = 4x. exp[x2]+8%.exp(x2]+8x3exp [x2] = 12x. exp(x2]+8x3. exp [x2] Evaluation of those functions at x = 0 gives f (0 ) = exp [0] = 1 f' (0) = 0 f" (0) = 2 f " ( 0 ) = 0 Therefore the Taylor Series is exp [x2]=1+f(0) (x)+(0) (Az) +(o)(Az)+ Q(6x") exp (x2] = 1+ ( x) 2 + Q (0x+) 4. Find the Power series solution to ; We will assume a power series solution as n =0 y = { an x " = a+ + Q 1 x + azx + ... Plug it into the original equation gives odby = " an . n . x " = 1 . a1 + 29 , x + 303x'+ ... n = 1 = E any ( n + 1 ) x " n = 0 Since dy = y therefore 8 dx Zany (n+1) x" = = anx" n=D This means, the coefficients must be an an = an+1 (n+1) or On+1 = n+1 Therefore y = a0+ Q1x+azx2+a3x3+ ... =a0+ Q.x += x2 + az x 3+ .. = ao + Ge x + mix2 x2+ ao 23+ ... 1x 2x3 = Qo(1+x+2/2! +x3/3!+ ... ) = a0 . exp ( x )