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Fall 08

# Fall 08 - 2 Mathematics 263 Ordinary Differential Equations...

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Unformatted text preview: 2 Mathematics 263 Ordinary Differential Equations and Linear Algebra 1. Find the general solutions to the following ODE’s; 1n part (a), also ﬁnd the speciﬁc solution satisfying the given initial condition. . (a) (2111' — 22:)y’ :31;2 — 6y with y(0) ; 2, (b) (1 + \$2)(y’ + 33/) = e-3’z‘ . (c) (152:2 + Szy2 - 18y)d:c + (4:331 -— 62:)dy = 0 (d) 1011' + \$11; chi/3 2. Solve the following Euler equation. 29:21]" + 3311' — y = z‘l (z > 0). for 1r < z < E cos2z 2 2" i 4. For the ODE 11:23; ” —(3:l: + 22:3)y' + (3 + 2x2)-y = 0, verify that y1 = :1: is a solution. ‘ Find the general solution of the given ODE. '3. Using variation of pararneters,_-solve y” + 4y = 5. Solve the following initial 'value problem. (D3 - D)y = z +' 3c” with y_(0) = y’(.0) = _y"(0) = V 6. Find the inverse Laplace transforms of the following 1 (a) 17(3)»: (s4 _ 1)" (b) F(8)= W -3 (c) F(s)= s2 + 343 + 5 7. Solve y’ - 2y :2 f (t) + 6(t ~ 2) with y(0) = 0 where f (t) is given by 2t 0 g t < l f(t) -{ 2 1 s ' t 8. Put the following system into matrix form Dv =-Av where v =y ( 3:8 ) y; = 4111 - 31/2 11% ‘= 2111 ‘ 33’? (a) Display the matrix A and ﬁnd a basis of R2 consisting of eigenvectors of A. (b) Find the general solution of the system Dv- — Av. (c) Calculate e’“. 9. Find bases for the column space, null space (i. e. solutions of the corresponding homogeneous equations) and row space for the following matrix A 1 —2 3 2 0 ~ A = —3 . 6 ~8' 5 —.2 4 —8 11 —3 2 ...
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