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APM 2611- DIFFERENTIAL EQUATIONS-2019
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lOMoARcPSD|12263423APM2611/201/1/2019
APM 2611- DIFFERENTIAL EQUATIONS-2019
Tutorial letter 201/1/2019
DIFFERENTIAL EQUATIONS
APM2611
Semester 1
Department of Mathematical Sciences
This tutorial letter contains solutions for assignment 01.
Downloaded by Anna Maina (annamurugijoe@gmail.com)
, lOMoARcPSD|12263423
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Downloaded by Anna Maina (annamurugijoe@gmail.com)
, lOMoARcPSD|12263423
Question 1
(1.1) Find an interval of definition for the equation
dx
x2 = 3x3t2.
dt
Since the equation has the solution x(t) = 0 for all t, any interval may be taken as an interval
of definition. The largest interval of definition is (−∞, ∞).
(1.2) Solve the initial value problem
dx
x2 = 3x3t2, x(0) = 1
dt
and give an interval of definition.
Notice that x = 0 is a solution for the equation, but does not satisfy the initial value problem.
The equation is separable, separating variables yields
dx
= 3t2 dt.
x
Integrating both sides yields
ln |x| = t3 + c
where c is the constant of integration. Since x(0) = 1
ln |1| = 03 + c
we have c = 0. Thus
3
ln |x| = t3 ⇒ |x| = et .
Since x(0) = 1 (which is positive), and by continuity of x(t)
3
x = et .
Since the solution is valid for all t, an interval of definition is (−∞ , ∞ ) or any other interval
which contains the initial value 0.
2
Downloaded by Anna Maina (annamurugijoe@gmail.com)
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