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# handout_1 - ACM95b/100b Lecture Notes Niles A Pierce...

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ACM95b/100b Lecture Notes Niles A. Pierce Caltech 2010 Introduction to Diﬀerential Equations Diﬀerential equations relate derivatives of functions: a) Newton’s second law F = ma = m dv dt = m d 2 x dt 2 (1) b) Radio carbon dating dy dt = ky, k = constant < 0 (2) c) Charge on a capacitor for a circuit L d 2 Q dt 2 + R dQ dt + 1 C Q = E ( t ) (3) d) Inviscid compressible ﬂow over an airplane ∂W ∂t + ∂F ∂x + ∂G ∂y + ∂H ∂z = 0 (4) e) Population dynamics dx dt = ax - αxy prey (5) dy dt = - cy + γxy predator (6) Ordinary diﬀerential equations (ODEs) have one independent variable: d 2 u dt 2 + du dt + u = t 3 - 1 (7) Partial diﬀerential equations (PDEs) have more than one independent variable: 2 u ∂x 2 + 2 u ∂y 2 = 0 (8) Usually we are interested in solutions to the diﬀerential equation that satisfy additional constraints. a) Initial value problems (IVPs) have initial conditions speciﬁed at a single point: y

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handout_1 - ACM95b/100b Lecture Notes Niles A Pierce...

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