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 9.3.49a

Chapter 9, Problem 9.3.49a

{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.
a. Find the solution of the initial value problem.

Verified step by step guidance

Recognize that the differential equation given is \(h'(t) = -k \sqrt{h(t)}\), where \(h(t)\) represents the water depth at time \(t\), and \(k\) is a positive constant.

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 \(\int \frac{1}{\sqrt{h}} \, dh\) on the left and \(\int -k \, dt\) on the right. Recall that \(\int h^{-1/2} \, dh = 2 \sqrt{h}\).

After integration, you will have an equation involving \(\sqrt{h}\) and \(t\) plus an integration constant \(C\). Use the initial condition \(h(0) = H_m\) to solve for \(C\).

Finally, solve the resulting equation for \(h(t)\) explicitly to express the water depth as a function of time.

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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 outflow velocity is proportional to the square root of the fluid’s height above the hole. In this problem, it leads to the differential equation h'(t) = -k√h, relating the rate of change of water depth to the current depth.

Separable Differential Equations

A separable differential equation can be written as a product of a function of t and a function of h, allowing variables to be separated on opposite sides of the equation. Here, h'(t) = -k√h can be rearranged to integrate both sides and find an explicit solution for h(t).

Initial Value Problem (IVP)

An initial value problem involves solving a differential equation with a given initial condition, such as h(0) = Hm. This condition allows determination of the constant of integration after solving the differential equation, yielding a unique solution that models the water depth over time.

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Initial Value Problems

Related Practice

Textbook Question

[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.

a. Find the general solution of the equation and express it explicitly as a function of t in two cases: y > 0 and y < 0.

Textbook Question

Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.

a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).

Textbook Question

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

a. Find the general solution of the equation.

e⁻ʸᐟ²y'(x) = 4x sin x² − x; y(0) = 0, y(0) = ln(1/4), y(√(π/2)) = 0

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The direction field for the differential equation y′(t)=t+y(t) is plotted in the ty-plane.

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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.

where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.

a. Verify by substitution that the general solution of the equation is P(t) = K/(1 + Ce⁻ʳᵗ), where C is an arbitrary constant.

Textbook Question

Stirred tank reaction A 100-L tank is filled with pure water when an inflow pipe is opened and a sugar solution with a concentration of 20 gm/L flows into the tank at a rate of 0.5 L/min. The solution is thoroughly mixed and flows out of the tank at a rate of 0.5 L/min.

c. At what time does the mass of sugar reach 95% of its steady-state level?

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