

Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
This is a summary sheet for the first half of the module, solving first and second order ordinary differential equations.
Typology: Cheat Sheet
1 / 3
This page cannot be seen from the preview
Don't miss anything!
On special offer
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