Problem 3 - Calculus Early Transcendentals

Question

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \frac{\sqrt{1+x^{2}}}{1+y} \frac{d y}{d x}=-x $$

Step-by-Step Solution

Answer

The family of solutions is given by \( y = C' \cdot e^{-\sqrt{1+x^2}} - 1 \).

Step 1: Write the Given Differential Equation

We start with the given differential equation: \( \frac{\sqrt{1+x^{2}}}{1+y} \frac{d y}{d x}=-x \). Our goal is to separate the variables to solve the equation.

Step 2: Separate the Variables

The aim here is to have all terms involving \( y \) on one side and all terms involving \( x \) on the other side.Multiply both sides by \( (1+y) \) and \( dx \) to get:\[\sqrt{1+x^{2}} \, dy = -x(1+y) \, dx\]Now distribute the \( -x \):\[\sqrt{1+x^{2}} \, dy = -x - xy \, dx\]Rearrange to separate the variables:\[\frac{dy}{1+y} = -\frac{x}{\sqrt{1+x^2}} \, dx\]

Step 3: Integrate Both Sides

Integrate both sides of the separated equation.Integrating the left side:\[\int \frac{dy}{1+y} = \ln|1+y| + C_1\]Integrating the right side using substitution \((u = 1+x^2, \, du = 2x \, dx)\):\[\int \frac{-x}{\sqrt{1+x^2}} \, dx = -\int \frac{1}{2\sqrt{u}} du = -\sqrt{u} + C_2 = -\sqrt{1+x^2} + C_2\]Let \( C = C_2 - C_1 \).Equating, we get:\[\ln|1+y| = -\sqrt{1+x^2} + C\]

Step 4: Solve for y as an Explicit Function of x

To express \( y \) as an explicit function of \( x \), first solve for \( y \) from the equation \[\ln|1+y| = -\sqrt{1+x^2} + C\]Exponentiate both sides:\[|1+y| = e^{(-\sqrt{1+x^2} + C)} = e^{C} \cdot e^{-\sqrt{1+x^2}}\]Let \( C' = e^{C} \), so:\[|1+y| = C' \cdot e^{-\sqrt{1+x^2}}\]If \( 1+y = C' \cdot e^{-\sqrt{1+x^2}} \), then:\[ y = C' \cdot e^{-\sqrt{1+x^2}} - 1 \].

Key Concepts

Separation of Variables

Separation of variables is a powerful technique used to solve differential equations. It involves rearranging an equation so all terms containing the dependent variable are on one side and all terms containing the independent variable are on the other. This technique works particularly well with first-order differential equations, where the unknown function and its derivative are involved.
How to Use Separation of Variables:

• Identify the equation in the form \( \frac{dy}{dx} = g(x)h(y) \).
• Rearrange the equation to move all \( y \) terms to one side and all \( x \) terms to the other.
• Integrate both sides with respect to their respective variables.
• Solve the resulting expressions for the function \( y \).

This method simplifies the process of finding solutions by breaking down complex differential equations into parts that are easier to handle. The exercise demonstrates this technique by separating variables in the equation \( \frac{\sqrt{1+x^{2}}}{1+y} \frac{d y}{d x}=-x \). It rearranges to \( \frac{dy}{1+y} = -\frac{x}{\sqrt{1+x^2}} \ dx \), allowing separate integration of each side.

Integration by Substitution

Integration by substitution is a method used to simplify integration, making it possible to evaluate complex integrals. It involves changing variables to a simpler form, typically through an inspired guess of a new variable.
Steps for Integration by Substitution:

• Select a substitution: Choose a new variable, \( u \), that simplifies the integral's expression.
• Determine the derivative: Find \( du \), the differential of the substitution variable, in terms of \( dx \).
• Rewrite the integral: Express the integral in terms of \( u \) and \( du \), replacing the original variable.
• Integrate: Solve the simpler integral.
• Back-substitute: Convert back the integrated expression in terms of the original variable.

In our exercise, substitution is applied to integrate \( \int \frac{-x}{\sqrt{1+x^2}} \, dx \). By letting \( u = 1 + x^2 \), and noting that \( du = 2x \, dx \), the integral simplifies, making it more straightforward to solve.

Exponential Functions

Exponential functions are key in solving differential equations and often emerge as solutions where the rate of change of a quantity is proportional to its current value. An exponential function can take the form \( f(x) = ab^x \), where \( a \) is a constant and \( b \) is the base of the exponential function.
Characteristics of Exponential Functions:

• Continuous growth or decay depending on whether \( b > 1 \) or \( b < 1 \).
• The derivative of an exponential function is proportional to the function itself.
• Exponentials are involved in natural processes, including compounding interest, population growth, and radioactive decay.

In the solution, exponential functions appear when solving for \( y \). After using separation of variables and integration by substitution, we find \(|1+y| = Ce^{-\sqrt{1+x^2}}\). The e in \( e^{-\sqrt{1+x^2}} \) is the base of the natural logarithms, demonstrating the ubiquitous nature of exponential functions in differential equations.