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**Unformatted text preview: **1. [ 0 points ] Classify each of the following equations as (i) either ordinary or partial differential equations, (ii) either first, second or third order, (iii) either linear or nonlinear: ( a ) d 2 y dx 2 + sin( t + y ) = cos( t ) Solution: ordinary, 2nd-order, nonlinear ( b ) d 2 y dx 2 + y ln( t ) + 4 dy dx = e t Solution: ordinary, 2nd-order, linear 2. [ 0 points ] Determine whether each of the two functions is a solution to the given differential equation: ( a ) t 2 y 00 + 5 ty + 4 y = 0 , t > 0; y 1 ( t ) = t- 2 ln( t ) , y 2 ( t ) = 2 t- 2 Solution: [take derivatives, plug them in] both y 1 and y 2 solve the differential equation ( b ) 2 t 2 y 00 + 3 ty- y = 0 , t > 0; y 1 ( t ) = t 1 / 2 , y 2 ( t ) = 2 t- 1 Solution: [take derivatives, plug them in] both y 1 and y 2 solve the differential equation 3. [ 0 points ] Solve the following linear first-order initial value problem y- y = 2 te 2 t , y (0) = 1 Solution: Using the method of integrating factors (see chapter 2, B&D), we get y ( t ) = 2 te 2 t- 2 e 2 t + 3 e t 4. [ 0 points ] Solve the following separable first-order initial value problem y = x ( x 2 + 1) / 4 y 3 , y (0) =- 1 √ 2 Solution: Using the method for separable equations, or even for exact equations, we get the implicit solution given by y 4- 1 4 x 4- 1 2 x 2- 1 4 = 0 . If you want, you can instead solve for the explicit solution, y ( x ) =- p ( x 2 + 1) / 2 . 5. [ 0 points ] Sketch several solutions to the given autonomous differential equation in the t-y plane. Determine the critical (equilibrium) points, and classify each as asymptotically stable, unstable or semistable. dy dt = ( y 2- 9)( y 2- 4 y + 4) Solution: [Draw some pictures, see chapter 2 B&D]. The critical points are at x = 3 (unstable), x =- 3 (stable) and x = 2 (semistable). 6. [ 0 points ] Solve the following exact first-order ODE y =- 2 x + 3 y 3 x + 4 y Solution: Using the method for exact first-order differential equations (B&D chapter 2), the implicit solution is given by x 2 + 3 xy + 2 y 2 = c . Again, you can solve for the explicit solution if you wish, but it is unnecessary unless it is stated in the problem. 7.7....

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