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In Problems 13–32 use variation of parameters to solve the given nonhomogeneous system.
21.
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A First Course in Differential Equations with Modeling Applications (MindTap Course List)
- 1. Find the general solution of y"" - 2y" - y' + 2y = 0arrow_forward1. Find the solution to the initial value problem 4x3 + 1 2у — 6 y(1) = 2. A. y = 3 – Vxª + x – 1 B. y = 2+ Vx³ + x – 2 C. y = 1+ Vx4 + x – 1 D. y = 4 – V4x³ + x – 1 E. y = V4³ + x – 1arrow_forward7. A scientist places two strains of bacteria, X and Y, in a petri dish. Initially, there are 400 of X and 500 of Y. The two bacteria compete for food and space but do not feed on each other. If x = x(t) and y = y(t) are the numbers of strains at time t days, the growth rates of the two populations are given by the system x' = 1.2x – 0.2y, y' = -0.2x + 1.5y Determine what happens to these two populations by solving the system of differential equations.arrow_forward
- Solve each second-order IVP. 1. y" + 2y – 15y = 0, y(0) = 2, /(0) = -6 . 2. y" + 6y' + 13y = 0, y(0) = -1, y (0) = 5 3. y" + 2y +y = 0, y(0) = 3, y'(0) = –1arrow_forward4. Find the general solution of 3 + 3= 2x y A. 1/(x*y) = -x² + C B. 1/(x³y³) = -x? + C dx C. 1/(x'y*) = x² + C D. 1/(x*y*) = x² + C %3D %3Darrow_forwardSuppose we are given y1(x) and y2(x) (with y1 # y2), which are two different solutions of a nonhomogeneous equation y" + p(x)y + q(x)y = g(x) In three steps, describe how to write down the general solution of (1): (1) Step 1: Step 2: Step 3:arrow_forward
- Q. No. 11 The solution of the DE 3ry" + y/ – y = 0 (a) yı = rš[1 – {x +²+...], y2 = 1+x – 20² + ... (b) yı = a3[1 – r +a² + ...], y2 = 1+ 2x – 2x² + ... (c) yı = xš[1 – x + a² + ...], y2 =1+ 2x – 2x3 + ... (d) yı = [1 – x + x² + ...], y2 = 1+ 2x – 2x2 +... solve this and tick the correct optionarrow_forward1. Show that y = ze* + e-2* is a solution of y' + 2y = 2e*. 3arrow_forwardExample 10.34. Solve the equation y" = x + y with the boundary conditions y(0) = y(1) = 0.arrow_forward
- In Problems 1–8 use the method of undetermined coeffi cients to solve the given system. dx 1. = 2x + 3y – 7 dt dy 2у + 5 = -x - dt dx 2. dt 5х + 9у + 2 dy — —х + 11у +6 dt -G )x+ (,) 3 3. X' = 3 t (4t + 9er\ 4. X' = 4 -4 х+ 1 -t + e6t 3 X + le' 10 5. X' = sin t 5 X + 1 6. X' 1 -2 cos 7. X' = 2 3 X +| -1 \0 0 5, 2 8. X' = 5 0 ]X + - 10 5 0 0/ 40/arrow_forward1. The Lotka-Volterra or predator-prey equations dU = aU – UV, dt (1) AP = eyUV – BV. dt (2) have two fixed points (U., V.) = (0,0), (U., V.) = (- :). The trivial fixed point (0,0) is unstable since the prey population grows exponentially if it is initially small. Investigate the stability of the second fixed point (U..V.) = 6:27 PM 3/3/2021 近arrow_forwardConsider the following initial value problem: Edit 1 y" + 8y + 15y = 8(t – 5) + u10(t); y(0) = 0, y(0) = = 4 a) Find the solution y(t). ("-") 1 (e-St – e 5) y(t) = 8. 1 -3(t-5) 1 –5(t–5) ult) X 2 1 e-5(t-10) 10 1 -3(t-10) 6. Ud(t) 15 where c = 5 and d 10arrow_forward
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