Mathematical Tools for Physics - Department of Physics - University

4—Differential Equations 74 Examples of equations that show up in physics problems are y′′+y= 0 (1−x^2 )y′′− 2 xy′+(+ 1)y= 0 reg ...

4—Differential Equations 75 Let=kin the first sum. Let=k+ 2in the second. Explicitlyshow the limits of the index on the sums, or ...

4—Differential Equations 76 This series looks suspiciously like the series for the sine function, but is has some of thex’s or s ...

4—Differential Equations 77 Some of the numerical methods you will find in chapter 11 start from the ideas of these expansions, ...

4—Differential Equations 78 4.6 Green’s Functions Is there a general way to find the solution to the whole harmonic oscillator i ...

4—Differential Equations 79 The way to make use of this picture is to take a sequence of contiguous steps. One step follows imme ...

4—Differential Equations 80 As a check for the plausibility of this result, look at the special case of small times. Use the pow ...

4—Differential Equations 81 Now integrate this to get L ∫ dI V 0 −IR =t+C, or − L R ln(V 0 −IR) =t+C Solve for the currentIto ge ...

4—Differential Equations 82 These are respectively V 0 eiωt=IR=I 0 eiωtR V 0 eiωt=q/C =⇒ iωV 0 eiωt=q/C ̇ =I/C=I 0 eiωt/C V 0 ei ...

4—Differential Equations 83 What happened to thee−Lt/Rterm of the previous solution? This impedance manipulation tells you thein ...

4—Differential Equations 84 Divide by the factor on the left side and you have x= ed−fb ad−bc , y= fa−ec ad−bc (4.44) provided t ...

4—Differential Equations 85 For whatever reason, I would like to get a non-zero solution forxandy. Can I? The condition depends ...

4—Differential Equations 86 The results of problem1.23and of section4.9tell you all about such equations. In particular, for the ...

4—Differential Equations 87 The two second order differential equations have four arbitrary constants in their solution. You can ...

4—Differential Equations 88 Assume a Frobenius solutions aboutx= 0 y= ∑∞ 0 akxk+s and substitute into (4.55). Could you use an o ...

4—Differential Equations 89 and the general solution is a sum of these. This procedure gives both solutions to the differential ...

4—Differential Equations 90 4.12 Asymptotic Behavior This is a slightly technical subject, but it will come up occasionally in e ...

4—Differential Equations 91 Even in the third case, the signs of the terms are the same after a while, so this is relevant to th ...

4—Differential Equations 92 2 These equations are separable, as in section4.7. Separate them and integrate, solving for the depe ...

4—Differential Equations 93 Problems 4.1 If the equilibrium positionx= 0for Eq. (4.4) is unstable instead of stable, this revers ...