Chapter Notes (2)

This method of doing separation of variables is

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Unformatted text preview: g variable. This method of doing separation of variables is similar in spirit to using the differential to do substitutions in integration problems. The manipulations are easier to carry out, though the mathematical reasoning behind them is somewhat obscured. 20 2 Differential Equations Example 2.6: Solve the initial value problem dy = ( y + 1) x , and y (0) = 4 . dx Solution: The first step is to produce the general solution of the differential equation. The ODE has separable variables. We separate the variables by dividing both sides by y + 1 and multiplying by dx . This leads to dy = x dx . y +1 We now integrate both sides, treating each side as if the variable on that side were an independent variable. ∫ The integral dy x2 = xdx = + C . y +1 ∫ 2 ∫ dy /( y + 1) is evaluated using the substitution u = y + 1 and yields ln y + 1 , ignoring the constant of integration. From this we obtain the implicit form of the solution. ln y + 1 = x2 +C . 2 We solve for y by exponentiation of both sides. This yields, (after writing C for eC ) y + 1 = Ce x 2 /2 . The solution y is unwrapped from the absolute value as we did in Example 2.5, producing the general solution y = Ce x 2 /2 −1 . Having found the general solution, we can solve the initial value problem. Substituting x = 0 and y = 4 in the last equation gives 4 = Ce0 − 1 = C − 1 , so C = 5 and the solution of the initial value problem is y = 5e x 2 /2 − 1 .! Example 2.7: Find the general solution of y (1) = 4 . dy = dx Solution: 21 2 y +1 and determine the solution for which x 2 Differential Equations We can separate the variables yielding dy dx = . 2y +1 x We now integrate both sides. ∫ dy dx =∫ x 2 y +1 (2.3) On the left side of (2.3) we have ∫ dy = ∫ (2 y + 1)−1/ 2 dy . 2 y +1 Making the substitution u = 2 y + 1, du = 2 dy the last integral becomes 1 −1/ 2 1/ 2 ∫ u du = u = 2 y + 1 2 (ignoring the constant of integration). Therefore (2.3) evaluates to 2 y + 1 = ln x + C , (2.4) which is the implicit...
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