In Problems 21–24, solve the given initial value problem using the method of Laplace transforms. Sketch the graph of the solution.
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Fundamentals of Differential Equations and Boundary Value Problems
- Problem A.6 In parts A.6.a and A.6.b, re-arrange the equation so that it is in form 1, if possible. If it is not possible to put it in form 1, then put it in form 2, and find the integrating factor eJ P(x)dx_ Form 1: v(y)dy = w(x)dx dy *+ p(x)y = f(x) Form 2: dx The symbols a, b and c are nonzero constants. Your final answer must have like terms combined and fractions reduced. Also, your final answer is to have as few exponents as possible (an exponent that has more than one term is still a single exponent. For example, x³x2Dx-a, which has 3 exponents, should be re-expressed as x3+2b-a. which now has only 1 exponent). A.6.a. xa-le-In(x²)dx + dy = dx + 42 dx – xc-1eln(x?)dx r-a А.6.b. — 4dy + ydx dy – 7e-In(y")y²dxarrow_forwardQuestion No. 4: x²y' + xy = 1arrow_forwardProblem 7.4: Find the PS of (x' = 4x - 3y y' = 3x + 4y x(0) = 2 y(0) = 1arrow_forward
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- Solve the initial-value problems using Laplace transformation. Write complete solutions. 1. Df− f= 0; f(0) = -3 2. D2f− f = e2x; f(0) = Df(0) = 0 Here are the example that might help.arrow_forwardProblem B.5 The function y, = e2x is a solution to y" – 4y' + 4y = 0. This second-order ODE can be reduced to the first- order ODE w' = 0. Find a second linearly independent solution y2. Also, obtain the general solution. Do not use the textbook's formula 5 (shown in Theorem 3.2.1).arrow_forwardProblem. 9: Let z = x? 7 xy + 6 y? and suppose that (x, y) changes from (2, 1) to (1.95, 1.05 ). (Round your answers to four decimal places.) (a) Compute Az. (b) Compute dz. ?arrow_forward
- Example 1: A particle is moves along a coordinate axis in the positive direction to the right. Its position at time t' is given by s(t) = t³ – 4t + 2. Find v(1) and a 1), use these to answer the following questions: (a). Is the particle moving from left to right or from right to left at t =1? (b). Is the particle speeding up or slowing down at time t 1? Solution:arrow_forwardExample 10.11. Using modified Euler's method, find y(0.2) and y(0.4) given y = y + e*, y(0) = 0.arrow_forward10. Use the Laplace transform to solve the system 1 - 4r +y 0. where r(0) = 2 and y(0) = 5 %3Darrow_forward
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