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Chapter 7: Problem 7

Explain how to solve a separable differential equation of the form \(g(y) y^{\prime}(t)=h(t)\).

Short Answer

Expert verified
Answer: To solve the separable differential equation \(g(y)y'(t) = h(t)\), follow these steps: 1. Separate the variables: Rewrite the equation with only terms involving y on one side and terms involving t on the other side. 2. Integrate both sides with respect to their respective variables. 3. Solve for y(t), if possible.

Step by step solution

01

Separate the variables

Divide both sides by g(y) and then multiply both sides by dt to separate the variables: $$\frac{1}{g(y)} y'(t) \, dt = \frac{h(t)}{g(y)} \, dt$$
02

Integrate both sides

Integrate the left side with respect to y and the right side with respect to t: $$\int \frac{1}{g(y)} \, dy = \int \frac{h(t)}{g(y)} \, dt$$ Let's suppose that the integrals are called F(y) and G(t): $$F(y) = G(t) + C$$ Where C is an integration constant.
03

Solve for y(t)

Now, we'll solve the equation for y as a function of t, if possible. This step will depend on the specific functions g(y) and h(t), and might require further algebraic manipulation or integration techniques. Once y(t) is found, the separable differential equation will be solved.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integrating Factors
Integrating factors are an essential tool for solving differential equations that are not readily separable or linear. These are functions we multiply on both sides of the equation to convert it into an exact or easily integrable form. Although our separable differential equation doesn't require an integrating factor, it's beneficial to understand the concept as it opens doors to solving more complex differential equations.

Imagine a scenario where we have a non-separable, linear first-order differential equation in the form \( dy/dx + P(x)y = Q(x) \). An integrating factor, often denoted as \( \mu(x) \), would be a function of x such that when multiplied through, the left-hand side becomes the exact derivative of a product, e.g. \( \mu(x)P(x)y = (\mu(x)y)' \). To find \( \mu(x) \), we would integrate \( P(x) \) with respect to x.
Integration Techniques
When it comes to solving separable differential equations, integration is the star of the show. After variable separation, which aligns all ys on one side and all ts on the other, we must integrate to find the solution. However, depending on the function, various integration techniques may be used, such as substitution, integration by parts, partial fraction decomposition, or even numerical methods for more complex integrals.

Some solutions, \( F(y) \) for example, will come easily with basic antiderivatives. Others, like \( G(t) \) even after separation, may require one of the specialized techniques. It is crucial to recognize which method to apply to effectively integrate both sides of the equation.
Differential Equation Solutions
The essence of solving a differential equation is to find a function or a set of functions that satisfy the relationship expressed by the equation. For first-order separable differential equations, the general solution typically features an unknown integration constant, denoted by C. This constant is determined if an initial condition is provided.

For instance, once integrated, we would express the solution as \( F(y) = G(t) + C \), where \( F(y) \) and \( G(t) \) are the respective antiderivatives. We find the particular solution by plugging in the initial condition and solving for C. Thereafter, we solve for \( y(t) \) explicitly if possible to articulate the solution in terms of y solely as a function of t.
Variable Separation
Variable separation is a strategic approach to solving separable differential equations. The goal is to rearrange the equation so that all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the other side. When successfully separated, the equation can be expressed in a form that allows for integration with respect to each variable.

In our exercise, variable separation involves dividing by \( g(y) \) and multiplying the entire equation by dt, enabling us to have \( dy/g(y) \) on one side and \( h(t)dt/g(y) \) on the other. This sets the scene for independent integration with respect to y and t, which is the crucial step that leads us towards the solution to the differential equation.

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Most popular questions from this chapter

The work required to launch an object from the surface of Earth to outer space is given by \(W=\int_{R}^{\infty} F(x) d x,\) where \(R=6370 \mathrm{km}\) is the approximate radius of Earth, \(F(x)=G M m / x^{2}\) is the gravitational force between Earth and the object, \(G\) is the gravitational constant, \(M\) is the mass of Earth, \(m\) is the mass of the object, and \(G M=4 \times 10^{14} \mathrm{m}^{3} / \mathrm{s}^{2}\) a. Find the work required to launch an object in terms of \(m\) b. What escape velocity \(v_{e}\) is required to give the object a kinetic energy \(\frac{1}{2} m v_{e}^{2}\) equal to \(W ?\) c. The French scientist Laplace anticipated the existence of black holes in the 18 th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, \(c=300,000 \mathrm{km} / \mathrm{s},\) then light cannot escape the body and it cannot be seen. Show that such a body has a radius \(R \leq 2 G M / c^{2}\). For Earth to be a black hole, what would its radius need to be?

Compute the absolute and relative errors in using c to approximate \(x\). \(x=\pi ; c=3.14\)

Determine whether the following statements are true and give an explanation or counterexample. a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function. b. If the number of subintervals used in the Midpoint Rule is increased by a factor of \(3,\) the error is expected to decrease by a factor of 8. c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of \(4,\) the error is expected to decrease by a factor of 16.

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