Differential Equations Cheatsheet, Study notes of Differential Equations

Download Differential Equations Cheatsheet and more Study notes Differential Equations in PDF only on Docsity! Differential Equations Cheatsheet JMC Year 1, 2017/2018 syllabus Fawaz Shah Topics not covered in this summary: phase portraits, similarity transformations. Contents 1 Definitions 2 2 1st order linear ODEs 2 3 1st order non-linear ODEs 3 3.1 Exact equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Separable ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.3 Homogenous ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.4 Bernoulli type ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 2nd order ODEs 4 4.1 Special case - y missing . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.2 Special case - x missing . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 General case - finding the CF . . . . . . . . . . . . . . . . . . . . . 5 4.4 General case - finding the PI . . . . . . . . . . . . . . . . . . . . . . 6 5 Solving systems of differential equations 7 1 1 Definitions Order (of derivative) An nth derivative has order n. Order (of ODE) The order of the highest derivative present in an ODE. Degree (of ODE) The highest power to which a term is raised in an ODE (ex- cluding fractional powers). Linear (ODE) An ODE which has no terms raised to more than the 1st power, and with no y, x or other derivative terms multiplied by each other. System of diff. equations A set of simultaneous equations of derivatives, where derivatives of y, x etc. are given w.r.t. a parameter t Order (of system) The order of the highest derivative present in the system. Degree (of system) The highest power to which a term is raised in an ODE (excluding fractional powers). Linear (system) A system which has no terms raised to more than the 1st power, and with no y or other derivative terms multiplied by each other. Homogeneous (system) A system with no explicit functions of t (i.e. f(t)) present. 2 1st order linear ODEs Every 1st order linear ODE can be expressed as: dy dx + p(x)y = q(x) (1) These can ALL be solved by the integrating factor method: 1. Multiply both sides by exp( ∫ p(x) dx) 2. Use the reverse product rule to express the LHS as a single derivative (of a function of y). 3. Integrate both sides and rearrange. 2 4.2 Special case - x missing If we can write the 2nd derivative as: d2y dx2 = f(y, dy dx ) (14) (i.e. no x terms present), then we can make the same substitution. Let P = dy dx . This means d2y dx2 = dP dx , therefore we have: dP dx = f(y, P ) (15) However, this is not yet a 1st order equation since the derivative is w.r.t. x, but we only have y terms on the RHS. DIFFERENT TO LAST TIME: we must rewrite dP dx as a derivative with respect to y. Luckily, we can see that: dP dx = dP dy dy dx = P dP dy (16) Therefore: P dP dy = f(y, P ) (17) This is 1st order w.r.t P and can be solved by appropriate 1st order methods. 4.3 General case - finding the CF The general solution (GS) of a 2nd order ODE can be expressed as the sum of two other functions, called the ’complementary function’ (CF) and a ’particular integral’ (PI). yGS = yCF + yPI (18) A 2nd order ODE will usually be presented to us in the form: a d2y dx2 + b dy dx + c = f(x) (19) It can be shown that the CF can be calculated from the LHS of the above equation. We write down the auxiliary equation, which is simply the equation: aλ2 + bλ+ c = 0 (20) using a, b, c from above. Solving this gives us two values, λ1 and λ2. 5 Case 1: λ1 6= λ2, both real We can express the CF as: yCF = A1e λ1x + A2e λ2x (21) where A1 and A2 are arbitrary constants. Case 2: λ1 = λ2, both real Same as above, but we stick an x in front of one of the clashing parts of the solution. yCF = A1e λ1x + A2xe λ2x (22) Case 3: λ1, λ2 are complex If the auxiliary equation has complex roots, λ1 and λ2 will be complex conjugates. The CF can be expressed as: yCF = A1e (a+bi)x + A2e (a−bi)x = ea(A1e i(bx) + A2e −i(bx)) = ea(C1cos(bx) + C2sin(bx)) (23) where C1 = A1 +A2 and C2 = (A1−A2)i. Note that even though A1 and A2 may have been complex, C1 and C2 are necessarily real. 4.4 General case - finding the PI The particular integral is any function yPI that satisfies the ENTIRE differential equation. The particular integral can be calculated depending on the form of the RHS of equation 19. We will refer to the RHS as simply f(x) and the particular integral (as before) as yPI . We can follow some basic rules: Case 1: f(x) is a polynomial Try setting yPI as a general polynomial of the same degree. e.g. if f(x) is a quadratic, try setting yPI = ax2 + bx+ c and substituting into the ODE. We will solve for a, b, c, and this will give us yPI . Case 2: f(x) is a multiple of ebx, ebx NOT in CF Choose yPI = Aebx for some real number A. 6 Case 3: f(x) is a multiple of ebx, ebx IS in CF We now have a clash between the PI and the CF. We can try yPI = Axebx, i.e. sticking an x in the PI to avoid the clash. If this doesn’t work, we can choose yPI = A(x)ebx for some real FUNCTION A. Remember to use the CHAIN RULE to differentiate A this time. At the end remove any clashing terms, i.e. terms of the form Beλx where eλx is already present in the CF. Other terms with more x’s included are allowed, e.g. xeλx would not count as a clashing term. Case 4: f(x) = A(x)ebx where A(x) is a polynomial Choose yPI = C(x)ebx for some polynomial C(x). Case 5: f(x) is trigonometric (e.g. sin, cos, sinh etc.) Look for a pattern in f(x). A good tip for an f(x) with only sines/cosines is to use yPI = A cos(x)+B sin(x) and solve for A and B. A similar story for sinh and cosh. CAUTION: sinh, cosh and tanh are actually exponential functions in disguise, so make sure they do not clash with any eλx terms in the CF. Other cases If f(x) has a term of the form ex cos(x) or ex sin(x) then we can rewrite it as the real/imaginary part of a complex function (in this case e(1+i)x would be appropri- ate, since it expands to ex(cos(x) + i sin(x)). If f(x) is more complicated, we may have to be imaginative with the choice of yPI . e.g. for f(x) = Aeax + Bebx we could choose yPI = Ceax + Debx for some constants C,D. Again be careful of terms that clash with the CF. 5 Solving systems of differential equations A homogeneous 1st order system of equations can be written as: dx dt = F (x, y) dy dt = G(x, y) (24) 7