In Problems 1 and 2, fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol c₁ and has the form dy/dx = f(x, y). The symbols c₁ and k represent constants.

In Problems 3 and 4, fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c₁ and c₂ and has the form F(y, y″) = 0. The symbols c₁, c₂, and k represent constants.

In Problems 5 and 6, compute y′ and y″ and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c₁ and c₂ and has the form F(y, y′, y″) = 0. The symbols c₁ and c₂ represent constants.

In Problems 7–12, match each of the given differential equations with one or more of these solutions:

(a) y = 0, (b) y = 2, (c) y = 2x, (d) y = 2x².

7. xy′ = 2y
8. y′ = 2
9. y′ = 2y − 4
10. xy′ = y
11. y″ + 9y = 18
12. xy″ − y′ = 0

In Problems 13 and 14, determine by inspection at least one solution of the given differential equation.

13. y″ = y′
14. y′ = y(y − 3)

In Problems 15 and 16, interpret each statement as a differential equation.

15. On the graph of y = (x), the slope of the tangent line at a point P(x, y) is the square of the distance from P(x, y) to the origin.
16. On the graph of y = (x), the rate at which the slope changes with respect to x at a point P(x, y) is the negative of the slope of the tangent line at P(x, y).
17. (a) Give the domain of the function y = x^2/3.
    (b) Give the largest interval I of definition over which y = x^2/3 is a solution of the differential equation 3xy′ − 2y = 0.
18. 1. Verify that the one-parameter family y² − 2y = x² − x + c is an implicit solution of the differential equation (2y − 2)y′ = 2x − 1.
    2. Find a member of the one-parameter family in part (a) that satisfies the initial condition y(0) = 1.
    3. Use your result in part (b) to find an explicit function y = (x) that satisfies y(0) = 1. Give the domain of . Is y = (x) a solution of the initial-value problem? If so, give its interval I of definition; if not, explain.
19. Given that is a solution of the DE xy′ + y = 2x. Find x₀ and the largest interval I for which y(x) is a solution of the IVP
    

    ,

20. Suppose that y(x) denotes a solution of the initial-value problem y′ = x² + y², y(1) = −1 and that y(x) possesses at least a second derivative at x = 1. In some neighborhood of x = 1, use the DE to determine whether y(x) is increasing or decreasing, and whether the graph y(x) is concave up or concave down.
21. A differential equation may possess more than one family of solutions.
    1. Plot different members of the families y = ₁(x) = x² + c₁ and y = ₂(x) = −x² + c₂.
    2. Verify that y = ₁(x) and y = ₂(x) are two solutions of the nonlinear first-order differential equation (y′)² = 4x².
    3. Construct a piecewise-defined function that is a solution of the nonlinear DE in part (b) but is not a member of either family of solutions in part (a).
22. What is the slope of the tangent line to the graph of the solution of y′ = that passes through (−1, 4)?

In Problems 23–26, verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution.

23. y″ + y = 2 cos x − 2 sin x; y = x sin x + x cos x
24. y″ + y = sec x; y = x sin x + (cos x) ln(cos x)
25. x²y″ + xy′ + y = 0; y = sin(ln x)
26. x²y″ + xy′ + y = sec(ln x);
    

    y = cos(ln x) ln(cos(ln x)) + (ln x) sin(ln x)

In Problems 27–30, use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.

In Problems 31–34, verify that the indicated expression is an implicit solution of the given differential equation.

31. 
32. 
33. y″ = 2y(y′)³; y³ + 3y = 2 − 3x
34. (1 + xy)y′ + y² = 0; y = e−^xy
35. Find a constant c₁ such that y = c₁ + cos 3x is a solution of the differential equation y″ + 9y = 5.
36. Find constants c₁ and c₂ such that y = c₁ + c₂x is a solution of the differential equation y′ + 2y = 3x.
37. If c is an arbitrary constant, find a first-order differential equation for which y = ce^−x + 4x − 6 is a solution. [Hint: Differentiate and eliminate c between the two equations.]
38. Find a function y = f(x) whose graph passes through (0, 0) and whose slope at any point (x, y) in the xy-plane is 6 − 2x.

In Problems 39–42, is a two-parameter family of the second-order differential equation Find a solution of the second-order initial-value problem consisting of this differential equation and the given initial conditions.

In Problems 43 and 44, verify that the function defined by the definite integral is a particular solution of the given differential equation. In both problems, use Leibniz’s rule for the derivative of an integral:

43. y″ + 9y = f(x); f(t) sin 3(x − t) dt
44. [Hint: After computing use integration by parts with respect to t.]
45. The graph of a solution of a second-order initial-value problem d²y/dx² = f(x, y, y′), y(2) = y₀, y′(2) = y₁, is given in FIGURE 1.R.1. Use the graph to estimate the values of y₀ and y₁.
    

    

46. A tank in the form of a right-circular cylinder of radius 2 ft and height 10 ft is standing on end. If the tank is initially full of water, and water leaks from a circular hole of radius in. at its bottom, determine a differential equation for the height h of the water at time t. Ignore friction and contraction of water at the hole.
47. A uniform 10-foot-long heavy rope is coiled loosely on the ground. As shown in FIGURE 1.R.2 one end of the rope is pulled vertically upward by means of a constant force of 5 lb. The rope weighs 1 lb/ft. Use Newton’s second law in the form given in (17) in Exercises 1.3 to determine a differential equation for the height x(t) of the end above ground level at time t. Assume that the positive direction is upward.