# A Modern Introduction to Differential Equations by F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester By F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester

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An inﬁnite family of solutions may be characterized by n constants (parameters). These arbitrary constants, if present, may be evaluated by imposing appropriate initial conditions (usually n of them, involving behavior of the solution at a single point of its domain) or boundary conditions (at two or more points). Solving a differential equation with initial conditions is referred to as solving an initial-value problem (IVP). Solving a differential equation with boundary conditions is referred to as solving a boundary-value problem (BVP).

Solve the initial value problem x˙ = x 2 , x(1) = 1. b. If the solution in part (a) is valid over an interval I, how large can I be? c. Use technology to draw the graph of the solution x(t) found in part (a). d. Solve the initial value problem x˙ = x 2 , x(0) = 0. 4. The equation cQ dQ =− dP 1 + cP is one model used to estimate the cost of national health insurance,2 where Q(P) represents the quantity of health services performed at price P, P represents the proportion of the total cost of health services that an individual pays directly (“out of pocket expenses,” or coinsurance), and c is a constant.

If α < β, explain what happens to e(α−β)k t as t → ∞ and show that x(t) → α as t → ∞. 3 = Qt 2 +2Q , Q(1) = 1 explicitly for Q(t) and state the interval for 10. Solve the initial value problem dQ dt +3t which the solution is valid. C 1. A police department forensics expert checks a gun by ﬁring a bullet into a bale of cotton. The friction force resulting from the passage of the bullet through the cotton causes the bullet to slow down at a rate proportional to the square root of its velocity. 1 second and penetrated 10 feet into the bale of cotton.