APPM212
Tutoriaal/Tutorial 6
2026
Die vrae handel oor eenhede 10-11 (5.2-5.3 Trench).
The questions are on units 10-11 (5.2-5.3 Trench).
’n Toets oor hierdie tutoriaal sal op 25/03/2026 geskryf word.
A test on this tutorial will be written on 25/03/2026.
1. Vind ’n fundementele oplossingsversameling vir die volgende DVs
Find a fundamental solution set for the following DEs
(a) y ′′ − 6y ′ + 8y = 0
(b) y ′′ + 8y ′ + 41y = 0
(c) y ′′ − 2y ′ + y = 0
(d) 4y ′′ + y = 0
2. Los die volgende differensiaalvergelykings en aanvangswaardeprobleme op:
Solve the following differential equations and initial/boundary values problems:
(a) 4y ′′ + y = 0
2
−y =0
(b) ddt2y − 2 dy
dt
2
(c) ddt2y + dy
+y =0
dt
(d) 2y ′′ + 5y ′ + 3y = 0, y(0) = 3, y ′ (0) = −4
(e) 4y ′′ − 4y ′ + y = 0, y(0) = 1, y ′ (0) = −1.5
(f) y ′′ + 16y = 0, y(π/4) = −3, y ′ (π/4) = 4
(g) y ′′ + 2y ′ + 2y = 0, y(0) = 2, y ′ (0) = 1
(h) 4y ′′ + y = 0, y(0) = 3, y(π) = −4
3. (a) Skryf die algemene vergelyking van ’n nie-homogene tweede orde lineêre differensiaalvergelyking neer.
Write down the general equation of a nonhomogenous second order linear differential
equation.
(b) Wanneer sal ’n nie-homogene lineêre AWP ’n unieke oplossing hê? Gee die stelling se
bewoording.
When does a non-homogenous second order linear IVP have a unique solution? State the
theorem.
4. Beskou nie-homogene tweede orde DV
Consider the non-homogeneous second order DE
y ′′ + y ′ − 2y = 2x
(a) Op watter interval sal hierdie BWP ’n unieke oplossing hê?
On which interval will the IVP has a unique solution?
(b) Bepaal twee lineêr onafhanklike oplossings y1 en y2 van die komplementêre vergelyking.
Find two linear independent solutions y1 and y2 of the complementary equation.
(c) Bepaal die algemene oplossing van die komplementêre vergelyking.
Find the general solution of the complementary equation.
(d) Bepaal ’n spesifieke oplossing van die nie-homogene DV.
Determine a particular solution of the non-homogeneous DE.
(e) Bepaal die algemene oplossing van die nie-homogene DV.
Determine the general solution of the non-homogeneous DE.
Kort antwoorde
Short answers
1. (a) {e4x , e2x }
(b) {e−4x cos 5x, e−4x sin 5x}
(c) {ex , xex }
(d) {cos(x/2), sin(x/2)}
2. (a) y = c1 cos( 21 x) + c2 sin( 12 x)
√
√
(b) y = c1 e(1+ 2)x + c2 e(1− 2)x
h
√ √ i
(c) y = e−x/2 c1 cos 23 x + c2 sin 23 x
(d) y = 2e−3x/2 + e−x
(e) y = ex/2 − 2xex/2
(f) y = 3 cos 4x − sin 4x
(g) e−x (2 cos x + 3 sin x)
(h) y = 3 cos(x/2) − 4 sin(x/2)
3. (a) y ′′ + p(x)y ′ + q(x)y = f (x), with f (x) ̸≡ 0
(b) Theorem 5.3.1
4. (a) R
(b) y1 = ex and y2 = e−2x
(c) yc = c1 ex + c2 e−2x
(d) yp = −x − 21
(e) y = c1 ex + c2 e−2x − x − 21