Math 33B Midterm 1 Review Sheet: First-Order and Second-Order Differential Equations, Exams of Differential Equations

Download Math 33B Midterm 1 Review Sheet: First-Order and Second-Order Differential Equations and more Exams Differential Equations in PDF only on Docsity! Math 33B - Midterm 1 Review Sheet 1. First-order differential equations • Separable equations: y′ = P (x) Q(y) Write as Q(y)dy = P (x)dx and integrate both sides. See Section 2.2, problems 1 - 18 • Linear equations: y′ + p(x)y = g(x) Make sure equation is in “standard form” first! Multiply whole equation by integrating factor u(x) = e R p(x)dx and integrate both sides. Left hand side becomes u(x)y. See Section 2.4, problems 1 - 21 • Exact differential forms: P (x, y)dx + Q(x, y)dy = 0 Try to find an integrating factor u(x, y) to make the differential form exact. Then find a potential function f(x, y), i.e. a function for which ∂f ∂x = P and ∂f ∂y = Q. The solution of the differential equation is then f(x, y) = C. – If the form is exact, then ∂P ∂y = ∂Q ∂x – The equation is separable if there’s some integrating factor that makes P a function of x alone and Q a function of y alone. – The equation is linear if, after you divide by Q(x, y), it has the form (p(x)y + g(x))dx + dy = 0. The integrating factor is then u(x) = e R p(x)dx. – If h = 1 Q (∂P ∂y − ∂Q ∂x ) is a function of x alone, then let u(x) = e R h(x)dx. – If k = 1 P (∂Q ∂x − ∂P ∂y ) is a function of y alone, then let u(y) = e R k(y)dy. See Section 2.6, problems 9 - 29 • Qualitative analysis of autonomous equations: y′ = F (y) – Find equilibrium points: set F (y) = 0 and solve. – Classify each equilibrium as stable/unstable. – Sketch solution curves, draw phase line, etc. See Section 2.9, problems 15 - 31 2. Second-order linear homogeneous differential equations • Given two linearly independent solutions y1 and y2, the general solution is y = C1y1 + C2y2. • Given one solution y1, you can find another by reduction of order : – Let y2 = vy1 for some function v(t). – Take first and second derivatives of y2 and plug in. The v terms cancel, leaving only v′ and v′′. – Letting u = v′ gives a separable first-order equation, which you can solve to find u.