DIFFERENTIAL EQUATIONS MA133 CHEAT SHEET
This is a summary sheet for the first half of
the module, solving first and second order ordinary differential
equations. We assume conditions for FTC hold. You should learn all
these techniques by heart, and practice, practice, practice!
First Order Differential Equations We consider the main
scenarios
Trivial Case (Section 1.1)
dx
dt =f(t)
By Fundamental Theorem of Calculus simply integrate both sides
with respect to t
x(t) = Zf(t)dt
Linear Non-homogeneous (Sections 1.3/1.4/1.5)
dx
dt +p(t)x=q(t)
Multiply both sides by an Integrating Factor
P(t) = exp (Rp(t)dt) so that
d
dt(P(t)x(t)) = P(t)q(t)
Then integrate so that
x(t) = P(t)−1Zt
P(s)q(s)ds +AP (t)−1
Separable Equations (Section 1.6)
dx
dt =f(x)g(t)
First look for constant solutions, i.e. where f(x) = 0. Then look
for non-constant solutions (so f(x) never zero) and ”divide both
sides by f(x), multiply both sides by dt and integrate”.
Zdx
f(x)=Zg(t)dt
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