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Differential Equations Cheat Sheet, Cheat Sheet of Mathematics

This is a summary sheet for the first half of the module, solving first and second order ordinary differential equations.

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Uploaded on 10/09/2020

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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
1
pf3
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DIFFERENTIAL EQUATIONS MA133 CHEAT SHEET

This is a summary sheet for the first half ofthe module, solving first and second order ordinary differential equations. We assume conditions for FTC hold. You should learn allthese 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) =^ ∫ f (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 (∫^ p(t)dt) so that d dt(P^ (t)x(t)) =^ P^ (t)q(t) Then integrate so that x(t) = P (t)−^1 ∫ t 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”. ∫ (^) dx f (x) =

∫ g(t)dt

Autonomous First Order ODEs (Section 1.9) dx dt =^ f^ (x) Look for fixed points x∗, which satisfy f (x∗) = 0, i.e. are points where dx dt = 0. A fixed point x∗ is stable if f ′(x∗) < 0 and unstable if f ′(x∗) > 0.

Second Order Ordinary Differential Equations With Constant Coefficients

ad dt^2 x 2 + bdx dt + cx = f (t) The solution consists of x(t) = xc(t) + xp(t) where xc(t), the complementary solution, solves the homogeneous case f (t) = 0 and xp(t), the particular integral, gives the f (t). The Complementary Solution Solves ad dt^2 x 2 + bdx dt + cx = 0 Find the roots to the auxiliary equations aλ^2 + bλ + c = 0 i.e. λ± = −b±√ 2 ba^2 −^4 acthen we have

  • Real roots k 1 , k 2 complementary solution is Aek^1 t^ + Bek^2 t
  • Repeated real root k complementary solution is Aekt^ + Btekt
  • Complex roots p ± iq complementary solution is ept(A sin(qt) + B cos(qt)) or Aept^ cos(qt − φ) The Particular Integral Functions to ”guess”: