EE3121 Differential Equation
HW 1-3 (300 points in total)
1. (20 points) (a) Solve the following DE and find the highest point on the curve y = y(x).
y' + 3y' + 2y = 0, y(0) = 1, y' (0) = 6
(b) Solve the following initial value problem (IVP).
3y''' + 2y'' = 0, y(0) = − 1, y' (0) = 0, y'' (0) = 1
(Note that it is a third order DE.)
2. (20 points) Let us consider the following DE with a general solution y = y(x).
(a) Find two complete solutions y1 (x) and y2 (x) of Eq. (1) that satisfy the following IVPs.
y1 (0) = 3, y'1 (0) = 1
y2 (0) = 0, y'2 (0) = 1
(b) Find the interaction point: i.e., the coordinate of y1 (x) = y2 (x).
3. (20 points) Solve the following DEs.
(b) Find the general solution y = y(x).
4. (20 points) Find the general solution of the DEs. For IVPs, determine the complete solution.
(a)
(b)
(c)
(d)
5. (20 points) Find a power series solution centered at x = 0, where y(0) = a0 , y ′ (0) = a1 .
(a)
(b)
6. (20 points) Find the general solutions of the following differential equations.
(a)
(b)
7. (20 points) Consider two tanks T1 and T2, each containing 24L of water and it is assumed that the water does not overflow. Initially, we dissolved N1 (0) amount of salt in T1 and N2 (0) in T2 . After connecting the tanks, the liquid is circulated at a different rate. Specifically, liquid is leaving from tank T1 at a rate of 8L per minute, whereas liquid is leaving from tank T2 at a rate of 2L per minute. Calculate the amount of salt at time t in each tank, i.e., N1 (t) in tank T1 and N2 (t) in tank T2, respectively. Please provide derivations to receive full marks.
8. (20 points) Convert the following DE into a first-order Matrix form.
(a)
(b)
(c)
9. (20 points) Use the Matrix exponential to solve the following initial value problem (IVP).
10. (20 points) Solve the following system of DEs.
11. (20 points) Find the general solution of the following system of DEs.
(a)
(b)
(c)
12. (20 points) Find the general solution of the following system of DEs using Substitution Method.
(a)
(b)
13. (20 points) Find the general solution of the following DEs using Laplace Transforms.
(a)
(b)
14. (20 points) Find the general solution of the following DEs using Laplace Transforms.
(a)
(b)
y'' + 2y' + y = 1, y(0) = 2, y' (0) = −2
(c)
y'' + y = sin(3t), y(0) = y' (0) = 0
15. (20 points) Find the solution of the following DEs using Laplace Transforms.
(a)
y′′ + 7y′ + 12y = 3e −2t, y(0) = y′ (0) = 0
(b)
y'' + 8y' + 15y = −4δ (t − 7), y(0) = y' (0) = 0
(c)
y'' + 8y' + 20y = 4δ (t − 3) − 12δ (t − 5), y(0) = −2, y' (0) = 6
(d)
x' = x + 2y, y' = 2x − 2y, x(0) = 1, y(0) = 0