MAT3700 Assignments 02 2026 Due 2026

UNIVERSITY OF SOUTH AFRICA (UNISA)
College of Science, Engineering and Technology



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MAT3700
Applied Mathematics III

Assignment 02 — 2026


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Module Code: MAT3700

Module Name: Applied Mathematics III

Assignment No.: 02

Due Date: 2026




Submitted in partial fulfilment of the requirements for MAT3700
at the University of South Africa.

, UNISA | MAT3700 Assignment 02 — 2026



Question 1: D-operator Methods [24]

Find the general solutions of the following differential equations using D-operator methods.


1.1 [8]


Solve:
(D2 − 4D + 4) y = e2x + e−2x


Step 1: Auxiliary equation.


m2 − 4m + 4 = 0 =⇒ (m − 2)2 = 0 =⇒ m = 2 (repeated)


Step 2: Complementary function (CF).

Because the root m = 2 repeats twice, the CF takes the form:


yc = (C1 + C2 x) e2x


Step 3: Particular integral (PI) for e2x .

Since m = 2 is a double root, the standard exponential shortcut fails. Applying the repeated-
root rule gives:
1 2x x2 2x x2 2x
yp1 = e = e = e
(D − 2)2 2! 2

Step 4: Particular integral (PI) for e−2x .

Here D → −2 does not coincide with any root, so the substitution is direct:

1 1 1 −2x
yp2 = e−2x = e−2x = e
(D − 2)2 (−2 − 2)2 16


Step 5: General solution.

Final Answer
x2 2x 1 −2x
y = (C1 + C2 x) e2x + e + e
2 16




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