Department of Mathematics & Computer Science Chulalongkorn University First Semester 2018 2301216 Linear Algebra and Differential Equations (ISE)
Example of Final Examination 1. Find a general solution for each of the following equations
1.1. y00+ 4y0+ 5y= 0.
1.2. y00+y0−6y= 0.
1.3. y000−3y00+ 3y0−y=tet+ 4.
2. Determine a suitable form forY if the method of undetermined coefficients is to be used for 2.1. y00−4y0+ 8y= (2t2−3t)e2tcos(2t) + (10t2−t−1)e2tsin(2t)
2.2. y(4)−y000−y00+y0 =t2+ 4 +tsint
3. Find the general solution for the differential equation ty00−(1 +t)y0+y=t2e2t, fort >0, if it is known that y(t) =et is a solution of the corresponding homogeneous equation.
4. Solve the non-homogeneous differential equation t2y00+ty0+y= sec(lnt) for t >0.
5. Find a fundamental set of solutions for the homogeneous differential equation y(4)−8y0 = 0.
6. Solve the following vector differential equations.
6.1. ~x0 =
−12 0 1 −12
~ x
6.2. ~x0 =
3 9
−1 −3
~ x.
7. Find the general solution (as real-valued solution) for each of the following vector differential equations.
7.1. ~x0 =
1 −1 5 −3
~ x
7.2. ~x0 =
−3 √
√ 2 2 −2
~x+ 1
−1
e−t
7.3. ~x0 =
1 −1 1 −1
~ x+
1/t 1/t
. 8. For each of the systems:
(a) Classified the equilibrium points (node, saddle, center, or spiral).
(b) Determine the stability of the equilibrium points (stable, asymptotically stable, or unstable).
(c) sketch some trajectories in the phase plane.
8.1. ~x0 =
−1 0 0 −1
~ x
8.2. ~x0 =
−1 −1 2 −1
~ x
2
2301216 Formula Sheet 9 (Reduction of order).
y2 =y1
Z e−Rp(t)dt y12 dt
10 (Variation of parameters).
Y(t) =−y1(t)
Z y2(t)f(t)
W(y1, y2)(t) dt+y2(t)
Z y1(t)f(t) W(y1, y2)(t) dt 11 (Variation of parameter for a system).
X(t) = Φ(t)~ Z
Φ(t)−1f~(t) dt
