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/) = e3’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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 Summer '09
 SidneyTrudeau
 Differential Equations, Equations

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