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
- Suppose solving an equation by Laplace transform results in 4 (s+ 18) Y(s) (s +9)² * Evaluate y(0).arrow_forwardQUESTION 3 3 Find a7 in the decomposition of a(t) into tangential and normal components if r(t)=(,1-3t ) and t =-1,arrow_forward(2) Solve the following initial valued problems using the method of Laplace transformationarrow_forward
- 3. Solve using the Laplace transform and then solving the system of equations Differential Equationsarrow_forwardSolve the equations in Exercises 1–5 by the method of undetermined coefficients. 1. y′′ - y′ - 2y = 20 cos x 2. y′′ - y = ex + x2 3. y′′ + y = 2x + 3ex 4. y′′ + 2y′ + y = 6 sin 2x 5. y′′ - y′ - 6y = e-x - 7 cos xarrow_forwardSuppose solving an equation by Laplace transform results in 3s 60 Y (s): s2 + 20 s + 100 s2 + 20 s + 100 Evaluate y(0).arrow_forward
- Suppose solving an equation by Laplace transform results in 6 s Y(s) = s2 + 64° Evaluate y(r).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_forwardSolve 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_forward
- Problem B.3 The function y, = e-3t is a solution to y" – 3y' – 18y = 0. This second-order ODE can be reduced to the first- order ODE w' – 9w = 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). %3Darrow_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 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_forward
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