S13 - 1.3 Initial Conditions; Initial-Value Problems As we...

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1.3 Initial Conditions; Initial-Value Problems As we noted in the preceding section, we can obtain a particular solution of an n th order differential equation simply by assigning speciFc values to the n constants in the general solution. However, in typical applications of differential equations you will be asked to Fnd a solution of a given equation that satisFes certain preassigned conditions. Example 1. ±ind a solution of y ± =3 x 2 - 2 x that passes through the point (1 , 3). SOLUTION In this case, we can Fnd the general solution by integrating: y = ± ( 3 x 2 - 2 x ) dx = x 3 - x 2 + C. The general solution is y = x 3 - x 2 + C . To Fnd a solution that passes through the point (2 , 6), we set x = 2 and y = 6 in the general solution and solve for C : 6=2 3 - 2 2 + C =8 - 4+ C which implies C =2 . Thus, y = x 3 - x 2 + 2 is a solution of the differential equation that satisFes the given condition. In fact, it is the only solution that satisFes the condition since the general solution represented all solutions of the equation and the constant C was uniquely determined. ± Example 2. ±ind a solution of x 2 y ±± - 2 xy ± +2 y =4 x 3 which passes through the point (1 , 4) with slope 2. SOLUTION As shown in Example 4 in the preceding section, the general solution of the differential equation is y = C 1 x 2 + C 2 x x 3 . Setting x = 1 and y = 4 in the general solution yields the equation C 1 + C 2 + 2 = 4 which implies C 1 + C 2 . The second condition, slope 2 at x = 1, is a condition on y ± . We want y ± (1) = 2. We calculate y ± : y ± C 1 x + C 2 +6 x 2 , and then set x = 1 and y ± = 2. This yields the equation 2 C 1 + C 2 + 6 = 2 which implies 2 C 1 + C 2 = - 4 . Now we solve the two equations simultaneously: 11
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C 1 + C 2 =2 2 C 1 + C 2 = - 4 We get: C 1 = - 6 ,C 2 = 8. A solution of the differential equation satisfying the two conditions is y ( x )= - 6 x 2 +8 x +2 x 3 . It will follow from our work in Chapter 3 that this is the only solution of the differential equation that satisFes the given conditions. ± INITIAL CONDITIONS Conditions such as those imposed on the solutions in Examples 1 and 2 are called initial conditions. This term originated with applications where processes are usually observed over time, starting with some initial state at time t =0 .
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S13 - 1.3 Initial Conditions; Initial-Value Problems As we...

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