Mathematical Tools for Physics
4—Differential Equations 93 ofx^1 cosxabout the origin? This is singular there, but it’s easy to write the answer anyway because ...
4—Differential Equations 94 This determinesa 2 in terms ofa 0 ; it determinesa 4 in terms ofa 2 etc. a`=−a`− 2 1 (`+s)^2 −n^2 , ...
4—Differential Equations 95 This series looks suspiciously like the series for the sine function, but is has some of thex’s or s ...
4—Differential Equations 96 What isc(x)^2 +s(x)^2? Differentiate this expression to get d dx [c(x)^2 +s(x)^2 ] = 2c(x)c′(x) + 2s ...
4—Differential Equations 97 x t′ impulse t When the external force is the sum of two terms, the total solution is the sum of the ...
4—Differential Equations 98 To complete this idea, the external force is the sum of a lot of terms, the force betweent 1 andt 2 ...
4—Differential Equations 99 4.6 Separation of Variables If you have a first order differential equation — I’ll be more specific ...
4—Differential Equations 100 Manipulate this into L dI dt =V 0 −IR, then L dI V 0 −IR =dt Now integrate this to get L ∫ dI V 0 − ...
4—Differential Equations 101 4.7 Simultaneous Equations What’s this doing in a chapter on differential equations? Patience. Solv ...
4—Differential Equations 102 3a. The determinant is zero and so are both numerators. In this case the two lines are not only par ...
4—Differential Equations 103 The second is just− 2 times the first, so it isn’t a separate equation. The family of solutions is ...
4—Differential Equations 104 When you plug this into the differential equations for the masses, all the factors ofeαtcancel, jus ...
4—Differential Equations 105 These are negative, and that’s what you should expect. There’s no damping and the springs provide r ...
4—Differential Equations 106 With a little thought (i.e. don’t plunge blindly ahead) you can solve these easily. A 1 =A 2 =A 3 = ...
4—Differential Equations 107 4.9 Legendre’s Equation This equation and its solutions appear when you solve electric and gravitat ...
4—Differential Equations 108 a 2 =a 0 s(s+ 1)−C (s+ 2)(s+ 1) , then a 4 =a 2 (s+ 2)(s+ 3)−C (s+ 4)(s+ 3) , etc. (38) This looks ...
4—Differential Equations 109 The numerator in Eq. ( 37 ) foran+2is[(n+s)(n+s+ 1)−C]. If this happen to equal zero for some value ...
4—Differential Equations 110 Problems 4.1 If the equilibrium positionx= 0for Eq. ( 4 ) is unstable instead of stable, this rever ...
4—Differential Equations 111 4.8 For the undamped harmonic oscillator apply an extra oscillating force so that the equation to s ...
4—Differential Equations 112 4.14 Check the algebra in the derivation of then= 0Bessel equation. Explicitly verify that the gene ...
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