# Introduction to Ordinary Differential Equations. Academic by Albert L Rabenstein PDF By Albert L Rabenstein

ISBN-10: 1483212793

ISBN-13: 9781483212791

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Example text

We also simplify the expression for yp so that it becomes ν = ^ι'+··· + θ ; . 107) Taking the second derivative, we have fp =(Ci/; +-- + cy;) + (C1>1/ + ... + cn>;). ios) We shall also require that Ci>i' + - + C B >; = 0. 109) Then the expression for y"p simplifies to y; = ciyfi + - + cyn. iio) Continuing in this way, we find that the (n — l)th derivative will be of the form if we require that y{ri) = cly + ajn-i) + ...

In the current through it. Using this fact, t Unless the change is infinite, as happens in some idealized situations. 141). If the initial charge on the capacitance is zero, then L/'(0) = £(0), or /,(0)= E(0) L ' (L144) This condition is to be interpreted as meaning that /'(0 + ) = £(0)/L. Let us now assume that the applied voltage has the constant value E0. 142) becomes d 2I 2 i - dl JA 1 . 145) In order to determine the current I(t), we must find the solution of this equation that satisfies the initial conditions 7(0) = 0 , /'(0)=^.

The differential equation of motion of the body is d2x dt dx dt —-Ï2 = mqy — c —- , or d2x c dx "71 + — = ™9~d? 138) The initial conditions are x(0)=0, x'(0)=0. 139) The differential equation is nonhomogeneous, but has constant coefficients. Its general solution is found to be c The solution that satisfies the initial conditions is 2 - ( " ) ■ c The velocity of the body is dx dì m Q n e „-(c/«)n We note that as t-* + oo, the velocity approaches the limiting value mg c Let us next consider some applications of differential equations to electric circuits.