Q5.Solve the following ODE problem using Laplace:a- ÿ − 6ỷ + 13y = 0, where y(0) = 0 and y(0) = -3b- x - 2x + 3x = f(t), where x(0) = 0, x(0) = 4 and f(t) is represented by thefollowing graph:f(t) t43210-1-2-300.511.522.533.5t4

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Answered: Q5.Solve the following ODE problem…

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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We now turn our attention to functions of two or more variables. For example, the formula f(x, y) = x²y defines a function that gives us one output for each pair (x, y) which we input. If we plug in x = 2 and y = 1, we get f(2, 3) = (2)“(1)° = 4 ·1 = 4. If we plug in x = 3 and y = - 1, we get f(3, – 1) = (3)(– 1)° = 9 · ( – 1) = – 9. And there are infinitely many more points of the form (x, y) which we could input into f. %3D | Using the same function f(x, y) defined above, evaluate the following expressions: f(1, 3) = f(2, 2) f(- 2, – 2) = 1 f(50, 0) =
C) Use the Chain rule to find (). (). For the following functions: (1) u=xy-¹ where: x=e-2r cost r and y=√2r cse csc-¹2r 1 1 (2) w= + + where: x= and y=t¹sect and 2 = √t 1
Find g(3), g ′(3), and g ″(3).
2. Consider the function y(x) = 5x over the interval [0, 10]. (a) If the points A(0, y(0)), B(5, y(5)), and C(10, y(10)) are on the curve represented by the above function, find the slope of the lines (secants) AB, and AC, respectively. (b) If x represents time and y represents the displacement of a particle moving along a line, what is the physical meaning of your results to part (a)? (c) Now consider a point D(10 – h, y(10 – h)) on the curve represented by the given function, find the slope of the line DC. (d) As h → 0, what is the limiting value of your result to part (c)? (e) What is its physical meaning of your result to part (d) if again x represents time and y represents the displacement?
C. ƒ′(1) = (3e³ – 2p¯³) In (5p¨² — *√p) -
1. For the relation wc + w3 = 3c+ 9, find an expression for dc/dw. 2. (a) Suppose for a certain product, P = 30x – 0.6x? – 100, where P is total profit (in dollars) from making and selling x units. By differentiating this equation with respect to time t, derive an equation that relates x, dx/dt and dP/dt.

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Q5.Solve the following ODE problem using Laplace: a- ÿ − 6ỷ + 13y = 0, where y(0) = 0 and y(0) = -3 b- x - 2x + 3x = f(t), where x(0) = 0, x(0) = 4 and f(t) is represented by the following graph: f(t) t 4 3 2 1 0 -1 -2 -3 0 0.5 1 1.5 2 2.5 3 3.5 t 4