Problem #2: LetL(y) = any)(x) + an-13y(n − 1)(x) ++ [₁ y(x) + y(x)where do, alL(y) 3ex cos x + 8xeºx (*)Suppose that it is known thatL[v₁(x)] = 11xg⁹xL[12(x)] = 4e⁹x sin.xL[y3(x)]Se⁹x cos xFind a particular solution to (*).a,, are fixed constants. Consider the nth order linear differential equationwhen y(x)55xe⁹xwhen y(x) = 40e⁹x cos xwhen y3(x)30e⁹x cos x + 150€⁹x sin.x

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Answered: Let L(y) = any)(x) + an−1 y − ¹)(x)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let CER, 20 € Domain(y), and k Range(y). Construct an example of a second order, ordinary differential equation and verify a solution using the following steps. (a) Choose a function y(x) that is twice differentiable. Compute y'(x) and y"(x). (b) Choose functions (they may be constant) a2(x) and a(z). Then compute f(x) = a₂(x)y"(x) + a₁(x)y'(x) (c) Show that y(x) +C is a solution of f(x) = a₂(x)y"(x) + a₁(x)y'(x). (d) For what value(s) of C does y(zo) = k? Choose ro and k to give an example of an initial value problem using the ODE constructed in part (b). Provide the solution of the IVP. (e) Use a similar method (or share your own method) to construct another second-order IVP that you can verify the solution of without using any ODE technques other than the FTOC. Make sure to include justification for the solution and verify it.
Solve the following differential equation y' (x) = y· cos(x) O ข (2) sin(x) + C O ข (r) CeCos(z) O y (x) esin(z) + C O y (z) Cesin(z)
A Bernoulli differential equation is one that takes the form y' + P(x)y = Q(x)y".
Put the differential equation p(t) = g(t) = ty' (t² + 5)y cos(t) + e3t into the form y' + p(t)y = g(t) and find p(t) and g(t). Y help (formulas) help (formulas)
Let f(t) be the balance in a savings account at the end of t years and suppose y = f(t) satisfies the differential equation y' = 0.04y – 12,000. (a)Suppose that after one year the balance is $250,000. Is the balance increasing or decreasing at that time? At what rate is it increasing or decreasing at that time? (b)Write the differential equation in the form y' = k(y – M). (c)Describe the differential equation in words. %3D ... (a) The balance is at a rate of $ per year. (b) y' = (c) Choose the correct answer below. A. The rate of change of the savings account balance is proportional to the balance at the end of t years and is always decreasing. B. The rate of change of the savings account balance is proportional to the difference between the balance at the end of t years and $300,000. C. The rate of change of the savings account balance is proportional to the balance at the end of t years and is always increasing. D. The differential equation gives the account balance at any time, t.
Let P(t) be the population (in millions) of a certain city t years after 2015, and suppose that P(t) satisfies the differential equation P'(t) = 0.04P(t), P(0) = 7. (a) Use the differential equation to determine how fast the population is growing when it reaches 10 million people. (b) Use the differential equation to determine the population size when it is growing at a rate of 600,000 people per year. (c) Find a formula for P(t). (a) Choose the correct process to find how fast the population is growing when it reaches 10 million people. A. Evaluate P'(t) = 0.04(10). B. Evaluate P'(t) = 0.04P(10). C. Solve 10 =0.04P(t) for P(t). D. Solve P'(10) = 0.04P(t) for P(t).
Let P(t) be the population (in millions) of a certain city t years after 2015, and suppose that P(t) satisfies the differential equation P'(t) = 0.02P(t), P(0) = 3. (a) Use the differential equation to determine how fast the population is growing when it reaches 6 million people. (b) Use the differential equation to determine the population size when it is growing at a rate of 800,000 people per year. (c) Find a formula for P(t). (a) Choose the correct process to find how fast the population is growing when it reaches 6 million people. A. Evaluate P'(t) = 0.02(6). B. Evaluate P'(t) = 0.02P(6). %3D C. Solve 6 = 0.02P(t) for P(t). D. Solve P'(6) = 0.02P(t) for P(t). %3D
Consider the function of two variables u (x,t) = 2e-t sin (2x) Select the option that gives a partial differential equation for which this function of two variables is a solution. A. (∂2u/∂x2) = 4(∂2u/∂t2) B. (∂2u/∂x2) = (1/4)(∂2u/∂t2) C. (∂2u/∂x2) = - (1/4)(∂2u/∂t2) D. (∂2u/∂x2) = -4 (∂u/∂t) E. (∂2u/∂x2) = 4 (∂u/∂t)
Determine whether the following ordinary differential equation cosh (3/y) - In(7y)y" = 11cos(4yx) - y³y¹ + 9x is linear or nonlinear by matching it with the differential equation an(x) dry dxn -+an-1(x) an-ly. + dxn-1 + a₁(x)- + ao(x)y= g(x) dx dy Linear Nonlinear
Solve the differential equation y(t) = y'' +9y = √(t2), y(0) = y'(0) = 0. , t 2

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