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1. Given the equation dx = f(x, y), in which f(x, y) is a function of two variables defined on a region in the xy- plane. (a) State the order of the differential equation and justify your answer (b) Define a solution to the differential equation above and state what constitute its general solution (c) Suppose y(x) = yo, how does one obtain a particular solution to the equation? (d) State the linear standard form of the differential equation above and justify your answer

1. Given the equation dx = f(x, y), in which f(x, y) is a function of two variables defined on a region in the xy- plane. (a) State the order of the differential equation and justify your answer (b) Define a solution to the differential equation above and state what constitute its general solution (c) Suppose y(x) = yo, how does one obtain a particular solution to the equation? (d) State the linear standard form of the differential equation above and justify your answer

Advanced Engineering Mathematics
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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1. Given the equation dx = f(x, y), in which f(x, y) is a function of two variables defined on a region in the xy- plane. (a) State the order of the differential equation and justify your answer (b) Define a solution to the differential equation above and state what constitute its general solution (c) Suppose y(x) = yo, how does one obtain a particular solution to the equation? (d) State the linear standard form of the differential equation above and justify your answer 2. State Picard's theorem and hence using its iteration scheme, find yn (x) for n=0₂ for y' = x - y, y(0) = 1 dy dx 3. Solve 2 + y = 3 using the initial condition (a) y(0) = 1, (b) y(0) = 5 4. Find the particular solution of 3xy' - y = lnx + 1, x > 0 satisfying y(1) = -2 5. Given that the particular solution of y" - y' = cosx + sinx has the form Yp = Acosx + Bsinx. Determine the coefficients A and B 6. Solve the nonhomogeneous equation y"+y' - 2y = xe* 7. Solve the Euler equation x2y" -5x y' +9y = 0 8. Using Laplace transforms, solve 9. Find the general solution to 10. Given the system of coupled d²y dt² d²y dx² - 3d - 4 = 3e²t, with y(0) = 1 and y(0) = 0 dt +x ordinary differential equations below dy + y = 0 by the power series method dx - dx₁ = x₁ + x₂ dt dx2 = 4x₁ + x₂ dt (a) Put the above system of coupled ordinary differential equations (E) into the metric form X = AX, where A is square matrix of order 2x 2 By considering the solution to (E) in the form X = (b) Determine the characteristic equation of A and hence (c) Calculate the Eigen values(2) and Eigen vectors (V) (d) Deduce the general solution of (E) = Vext
Transcribed Image Text:1. Given the equation dx = f(x, y), in which f(x, y) is a function of two variables defined on a region in the xy- plane. (a) State the order of the differential equation and justify your answer (b) Define a solution to the differential equation above and state what constitute its general solution (c) Suppose y(x) = yo, how does one obtain a particular solution to the equation? (d) State the linear standard form of the differential equation above and justify your answer 2. State Picard's theorem and hence using its iteration scheme, find yn (x) for n=0₂ for y' = x - y, y(0) = 1 dy dx 3. Solve 2 + y = 3 using the initial condition (a) y(0) = 1, (b) y(0) = 5 4. Find the particular solution of 3xy' - y = lnx + 1, x > 0 satisfying y(1) = -2 5. Given that the particular solution of y" - y' = cosx + sinx has the form Yp = Acosx + Bsinx. Determine the coefficients A and B 6. Solve the nonhomogeneous equation y"+y' - 2y = xe* 7. Solve the Euler equation x2y" -5x y' +9y = 0 8. Using Laplace transforms, solve 9. Find the general solution to 10. Given the system of coupled d²y dt² d²y dx² - 3d - 4 = 3e²t, with y(0) = 1 and y(0) = 0 dt +x ordinary differential equations below dy + y = 0 by the power series method dx - dx₁ = x₁ + x₂ dt dx2 = 4x₁ + x₂ dt (a) Put the above system of coupled ordinary differential equations (E) into the metric form X = AX, where A is square matrix of order 2x 2 By considering the solution to (E) in the form X = (b) Determine the characteristic equation of A and hence (c) Calculate the Eigen values(2) and Eigen vectors (V) (d) Deduce the general solution of (E) = Vext
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