Problem 4 - A First Course in Differential Equations with Modeling Applications

Question

The dependent variable \(y\) is missing in the given differential equation. Proceed as in Example 1 and solve the equation by using the substitution \(u=y^{\prime}\) $$y^{\prime \prime}=1+\left(y^{\prime}\right)^{2}$$

Step-by-Step Solution

Answer

The solution is \( y = -\ln|\cos(x + C)| + C_3 \).

Step 1: Substitute Variables

Use the substitution \( u = y' \), which implies \( y'' = u' \). By substituting these into the given differential equation \( y'' = 1 + (y')^2 \), it becomes \( u' = 1 + u^2 \).

Step 2: Separate Variables

Rearrange the equation \( u' = 1 + u^2 \) by separating the variables. This can be rewritten as \( \frac{{du}}{{dx}} = 1 + u^2 \). Separate the variables and rewrite as \( \frac{{du}}{{1 + u^2}} = dx \).

Step 3: Integrate Both Sides

Integrate both sides of the equation. The left side is \( \int \frac{{du}}{{1 + u^2}} \), which integrates to \( \arctan(u) + C_1 \). The right side integrates to \( x + C_2 \). Set the equations to be equal: \( \arctan(u) + C_1 = x + C_2 \).

Step 4: Solve for Original Variable

Solve for \( u \) by isolating it: \( \arctan(u) = x + C \) (where \( C = C_2 - C_1 \)). Thus, \( u = \tan(x + C) \), and recall that \( u = y' \).

Step 5: Integrate to Find \( y \)

Find \( y \) by integrating \( u = y' = \tan(x + C) \). Integrate to find \( y \): \( y = \int \tan(x + C) \, dx = -\ln|\cos(x + C)| + C_3 \).

Key Concepts

Substitution Method

The substitution method is a powerful technique used in solving differential equations. This method simplifies the problem by introducing a new variable to replace a complex expression or derivative. In the exercise, we use the substitution \( u = y' \), where \( y' \) is the first derivative of \( y \). This step transforms a more complicated equation into a simpler one.

This method helps in:

• Reducing the complexity of the differential equation
• Making it easier to separate variables
• Simplifying the integration process by using known forms

Once the substitution is made, the initial equation which was in terms of second derivatives reduces to first-order equations. After using substitution, one can then potentially separate variables to make the problem tractable for integration.

Second-Order Differential Equation

A second-order differential equation includes second derivatives or the highest derivative is the second order. In our case, we started with the second-order equation \( y'' = 1 + (y')^2 \).

These types of equations appear frequently in physics and engineering describing variables with respect to forces, accelerations, and other numerous fields.The key points to learn about second-order differential equations are:

• Recognizing when a substitution can simplify the problem
• Understanding the reduction of order technique, simplify to first-order equations for ease of solution
• Being familiar with boundary and initial conditions to solve for constants in the integrations

By using appropriate methods as demonstrated, second-order differential equations can often be solved by reducing them to a simpler form.

Integration Techniques

Integrating is the crucial step in solving differential equations, especially after reducing the order of the equation using substitution. In this exercise, we performed integration to find \( u \) and ultimately \( y \).

Here are some essential integration techniques applied:

• Separation of variables: It involves rearranging the equation to allow integration on both sides. For our task, we reached an integrable form: \( \int \frac{du}{1 + u^2} = \int dx \).
• Inverse trigonometric functions: The integration of \( \int \frac{du}{1 + u^2} \) is a standard form that results in \( \arctan(u) \). Knowing these standard integrals is vital.
• Technique for definite and indefinite integrals: Balancing the constants of integration and appropriately modifying them to simplify our answer.

Integrating step by step, handling constants carefully, and recalling that \( u = y' \) leads us to find the complete solution for \( y \) by a second integration, \( y = \int \tan(x + C) \, dx \). Understanding these techniques enhances proficiency in tackling differential equations.