Linear differential equations with constant coefficients - existence of solutions, Wronskian, independence of solutions, initial value problems for second-order equations - One Line Questions

1. What is the Wronskian of y1(x) = e^x and y2(x) = e^(-x)? — -2
2. What is the Wronskian of y1(x) = x*e^x and y2(x) = e^x? — 0
3. What is the Wronskian of y1(x) = cos(x) and y2(x) = sin(x)? — 1
4. The existence and uniqueness theorem for a second-order linear differential equation ay'' + by' + cy = f(x) guarantees a unique solution when: — a, b, c, and f(x) are continuous on an interval I, and a(x) != 0 on I, with given initial conditions y(x0) = y0 and y'(x0) = y1
5. What is a particular solution (yp) of ay'' + by' + cy = f(x)? — Any one solution that satisfies the non-homogeneous equation
6. What is the characteristic equation associated with the differential equation ay'' + by' + cy = 0? — ar^2 + br + c = 0
7. What is the general form of a second-order linear differential equation with constant coefficients? — ay'' + by' + cy = f(x)
8. Applying the initial conditions y(0) = 2, y'(0) = 0 to y(x) = c1*e^x + c2*e^(-x) gives the system: — c1 + c2 = 2, c1 - c2 = 0
9. Solving c1 + c2 = 2 and c1 - c2 = 0 yields: — c1 = 1, c2 = 1
10. Solving the system c1 + c2 = 1 and c1 + 2c2 = 0 yields which values for c1 and c2? — c1 = 2, c2 = -1
11. The existence and uniqueness theorem for second-order linear differential equations with constant coefficients requires the function f(x) to be: — Continuous
12. The condition for a second-order linear differential equation with constant coefficients to have real, distinct roots in its characteristic equation depends on the: — Discriminant of the characteristic quadratic equation (b^2 - 4ac)
13. If the roots of the characteristic equation are complex conjugates alpha ± i*beta, the functions e^((alpha+i*beta)x) and e^((alpha-i*beta)x) form a fundamental set of solutions. Which other pair also forms a fundamental set? — e^(alpha*x)*cos(beta*x) and e^(alpha*x)*sin(beta*x)
14. If the characteristic equation has real roots r1 and r2, the functions e^(r1*x) and e^(r2*x) are solutions. If r1 = r2 = r, then the linearly independent solutions are: — e^(r*x) and x*e^(r*x)
15. For the differential equation ay'' + by' + cy = f(x), what are 'a', 'b', and 'c' if the coefficients are constant? — Constants
16. The concept of the Wronskian is primarily used to determine: — Linear independence of solutions
17. If y1(x) and y2(x) are linearly independent solutions to a homogeneous linear differential equation, their Wronskian W(y1, y2)(x) is: — Never zero
18. The Wronskian of the fundamental set of solutions {y1(x), y2(x)} for the homogeneous equation ay'' + by' + cy = 0 is: — Non-zero for all x in the interval of definition
19. For an initial value problem y'' + p(x)y' + q(x)y = f(x) with y(x0) = y0 and y'(x0) = y1, a unique solution exists on an interval if: — p(x), q(x), and f(x) are continuous on the interval and the coefficient of y'' is non-zero
20. What are the roots of the characteristic equation r^2 + 2r + 5 = 0? — r = -1 ± 2i
21. What is the root of the characteristic equation r^2 + 4r + 4 = 0? — r = -2 (repeated)
22. The roots of r^2 - 1 = 0 are: — r = 1, r = -1
23. For the differential equation y'' - 3y' + 2y = 0, what are the roots of the characteristic equation r^2 - 3r + 2 = 0? — r = 1, r = 2
24. For the equation y'' + y = 0, the characteristic equation is r^2 + 1 = 0. The roots are: — r = i, r = -i
25. Consider the initial value problem: y'' - y = 0, y(0) = 2, y'(0) = 0. The characteristic equation is: — r^2 - 1 = 0
26. Consider the initial value problem: y'' - 3y' + 2y = 0, with y(0) = 1 and y'(0) = 0. What is the characteristic equation? — r^2 - 3r + 2 = 0
27. Consider the equation y'' + 2y' + 5y = 0. What is the characteristic equation? — r^2 + 2r + 5 = 0
28. Consider the equation y'' + 4y' + 4y = 0. What is the characteristic equation? — r^2 + 4r + 4 = 0
29. For a non-homogeneous linear differential equation ay'' + by' + cy = f(x), where f(x) is not identically zero, the general solution is the sum of: — The complementary solution and a particular solution
30. What is the complementary solution (yc) of ay'' + by' + cy = f(x)? — The general solution of the associated homogeneous equation ay'' + by' + cy = 0
31. If the Wronskian of two solutions y1(x) and y2(x) to a homogeneous linear differential equation is non-zero for at least one point in an interval, what can be concluded about these solutions? — They are linearly independent
32. If y1(x) and y2(x) are linearly independent solutions to ay'' + by' + cy = 0, what is their Wronskian W(y1, y2)(x)? — W(y1, y2)(x) != 0 for all x
33. If y1(x) and y2(x) are two solutions to ay'' + by' + cy = 0, and y1(x) is a non-trivial solution, then y2(x) is linearly dependent on y1(x) if and only if: — y2(x) = k * y1(x) for some constant k
34. For the initial value problem y'' - 3y' + 2y = 0, y(0) = 1, y'(0) = 0, which of the following are the correct initial conditions to determine c1 and c2? — y(0) = c1 + c2 = 1, y'(0) = c1 + 2c2 = 0
35. What is the general solution for y'' + 4y' + 4y = 0? — y(x) = (c1 + c2*x)*e^(-2x)
36. If the characteristic equation ar^2 + br + c = 0 has one repeated real root r, what is the general solution to ay'' + by' + cy = 0? — y(x) = (c1 + c2*x)*e^(r*x)
37. What is the specific solution to the initial value problem y'' - 3y' + 2y = 0, y(0) = 1, y'(0) = 0? — y(x) = 2*e^x - e^(2x)
38. The general solution for y'' + y = 0 is: — y(x) = c1*cos(x) + c2*sin(x)
39. If the characteristic equation ar^2 + br + c = 0 has two distinct real roots r1 and r2, what is the general solution to ay'' + by' + cy = 0? — y(x) = c1*e^(r1*x) + c2*e^(r2*x)
40. The general solution for y'' - y = 0 is: — y(x) = c1*e^x + c2*e^(-x)
41. What is the general solution for y'' - 3y' + 2y = 0? — y(x) = c1*e^x + c2*e^(2x)
42. Consider the initial value problem: y'' + y = 0, y(0) = 1, y'(0) = 0. The specific solution is: — y(x) = cos(x)
43. What is the general solution for y'' + 2y' + 5y = 0? — y(x) = e^(-x) * (c1*cos(2x) + c2*sin(2x))
44. If the characteristic equation ar^2 + br + c = 0 has two complex conjugate roots alpha ± i*beta, what is the general solution to ay'' + by' + cy = 0? — y(x) = e^(alpha*x) * (c1*cos(beta*x) + c2*sin(beta*x))
45. The specific solution to y'' - y = 0, y(0) = 2, y'(0) = 0 is: — y(x) = e^x + e^(-x)
46. For the homogeneous equation ay'' + by' + cy = 0, if y1(x) and y2(x) are two solutions, and their Wronskian W(y1, y2)(x) = 0 for all x in an interval, what does this imply? — y1 and y2 are linearly dependent
47. What is the Wronskian of two functions y1(x) and y2(x)? — y1(x)y2'(x) - y1'(x)y2(x)
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