Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.3.49b

Chapter 9, Problem 9.3.49b

{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm.
b. Find the solution in k=0.1the case that and H=0.5m.

Verified step by step guidance

Start with the given differential equation from Torricelli's law: $h'(t) = -k \sqrt{h(t)}$, where $k = 0.1$ and the initial condition is $h(0) = H = 0.5$ meters.

Rewrite the differential equation in separable form: $\frac{dh}{dt} = -k \sqrt{h}$ can be rearranged as $\frac{dh}{\sqrt{h}} = -k \, dt$.

Integrate both sides: integrate $\frac{1}{\sqrt{h}} \, dh$ on the left and integrate $-k \, dt$ on the right. Remember to include the constant of integration after integrating.

Solve the integral on the left side, which is $\int h^{-1/2} \, dh = 2 \sqrt{h}$, and the integral on the right side, which is $-k t + C$ where $C$ is the constant of integration.

Use the initial condition $h(0) = 0.5$ to solve for the constant $C$, then express $h(t)$ explicitly as a function of $t$ by isolating $h$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Torricelli’s Law

Torricelli’s law describes the speed of fluid flowing out of an orifice under gravity, stating that the rate of change of the water height h(t) is proportional to the square root of h(t). Mathematically, it is expressed as h'(t) = -k√h, where k depends on gravity and the tank and hole dimensions.

Separable Differential Equations

The given differential equation h'(t) = -k√h is separable, meaning variables h and t can be separated on opposite sides of the equation. This allows integration with respect to each variable independently to find the explicit solution h(t).

Initial Value Problem (IVP)

An initial value problem involves solving a differential equation with a given initial condition, here h(0) = H. This condition is used to determine the constant of integration after solving the differential equation, ensuring the solution fits the physical scenario.

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Initial Value Problems

Related Practice

Textbook Question

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.)

y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1

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Textbook Question

{Use of Tech} Intravenous drug dosing The amount of drug in the blood of a patient (in milligrams) administered via an intravenous line is governed by the initial value problem y’(t) = -0.02y + 3, y(0) = 0 where t is measured in hours.

b. What is the steady-state level of the drug?

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Textbook Question

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m'(t) + km(t) = I, where m(t) is the mass of the drug in the blood at time t ≥ 0, k is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate.

b. Graph the solution for I = 10 mg/hr and k = 0.05 hr⁻¹.

Textbook Question

A bad loan Consider a loan repayment plan described by the initial value problem

B'(t)=0.03B−600,B(0)=40,000,

where the amount borrowed is B(0)=\$40,000, the monthly payments are \$600, and B(t) is the unpaid balance in the loan.

b. What is the most that you can borrow under the terms of this loan without going further into debt each month?

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Textbook Question

{Use of Tech} Chemical rate equations Let y(t) be t he concentration of a substance in a chemical reaction (typical units are moles/liter). The change in the concentration, under appropriate conditions, is modeled by the equation dy/dt=-ky^n for t≥0, where k>0 is a rate constant and the positive integer n is the order of the reaction.

b. Solve the initial value problem for a second-order reaction (n=2) assuming y(0)=y0.

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Textbook Question

Convergence of Euler's method Suppose Euler's method is applied to the initial value problem y′(t) = ay, y(0) = 1, which has the exact solution y(t) = eᵃᵗ. For this exercise, let h denote the time step (rather than Δt). The grid points are then given by tₖ = kh. We let uₖ be the Euler approximation to the exact solution y(tₖ), for k = 0, 1, 2, ...

b. Show by substitution that uₖ = (1 + ah)ᵏ is a solution of the equations in part (a), for k = 0, 1, 2, ...