Problem 17 - Calculus Single Variable

Question

Solve the given differential equation. $$ \frac{d y}{d x}=x \sec (y) $$

Step-by-Step Solution

Answer

The general solution is \( |\sec(y) + \tan(y)| = Ce^{\frac{x^2}{2}} \).

Step 1: Understand the form of the differential equation

The given differential equation is \( \frac{d y}{d x} = x \sec(y) \). This is a first-order differential equation involving the derivative of \( y \) with respect to \( x \).

Step 2: Separate variables

Rearrange the equation to separate the variables \( y \) and \( x \). We can write this as \( \sec(y) \ dy = x \ dx \), placing all \( y \) terms on one side and \( x \) terms on the other.

Step 3: Integrate both sides

Integrate both sides separately: \[ \int \sec(y) \ dy = \int x \ dx \]The left side uses the integral of \( \sec(y) \), and the right side uses the integral of \( x \).

Step 4: Evaluate the integrals

Compute both integrals: - The integral of \( \sec(y) \) is \( \ln |\sec(y) + \tan(y)| + C_1 \).- The integral of \( x \) is \( \frac{x^2}{2} + C_2 \).Thus, \( \ln |\sec(y) + \tan(y)| + C_1 = \frac{x^2}{2} + C_2 \).

Step 5: Simplify and solve the equation

Combine constants and rewrite the equation:\[ \ln |\sec(y) + \tan(y)| = \frac{x^2}{2} + C \],where \( C = C_2 - C_1 \). Exponentiate to eliminate the logarithm:\[ |\sec(y) + \tan(y)| = e^{\frac{x^2}{2} + C} = Ce^{\frac{x^2}{2}} \].

Step 6: Conclude the general solution

Solve for \( y \) if necessary, but typically we express the solution in terms of \( \sec(y) + \tan(y) \). Thus, the general solution is \[ |\sec(y) + \tan(y)| = Ce^{\frac{x^2}{2}} \].

Key Concepts

Separation of Variables

When faced with a differential equation like \( \frac{dy}{dx} = x \sec(y) \), one of the primary techniques used to solve it is known as **separation of variables**. This approach involves rearranging the equation to isolate all terms involving one variable (such as \( y \)) on one side, and all terms featuring the other variable (like \( x \)) on the opposite side.
Let's follow the separation of variables process from the step-by-step solution. Initially, the goal is to write the equation in a form where you have all \( y \)-related terms together as in \( \sec(y) \ dy \), and all \( x \)-related terms grouped as \( x \ dx \). It's like unmixing the variables to untangle the equation's intricacies, making it solvable through integration.

• Bring \( \sec(y) \) and \( dy \) together on one side.
• Place \( x \) and \( dx \) on the other side.

By successfully separating the variables, we simplify the problem into an easier form that allows direct integration. This sets the stage for the next steps in solving the differential equation.

Integration

Once the variables are separated, the next key step in solving the differential equation is **integration**. Integration is like piecing together tiny fragments of change, or finding the original equation knowing its rate of change. For our differential equation, this means integrating both sides to unravel the general solution.
The integral on the left side, \( \int \sec(y) \ dy \), involves integrating the secant function. This results in \( \ln |\sec(y) + \tan(y)| + C_1 \) due to the nature of the secant's relation to the tangent function. Meanwhile, the integration of \( x \) on the right gives us \( \frac{x^2}{2} + C_2 \).

• The left side's integral uses properties and identities of trigonometric functions.
• The right side is a straightforward power rule integration application.

Through integration, you bridge the form of the separated equation to a more complete equation, which will eventually highlight the solution’s structure. This transformation paves the way to synthesizing a comprehensive look at how the function changes over its domain.

General Solution

The culmination of solving a differential equation is finding its **general solution**. A general solution helps us understand the breadth of possible answers that satisfy the differential equation. Post integration, we combined the constants into a single constant \( C \), yielding the equation \( \ln |\sec(y) + \tan(y)| = \frac{x^2}{2} + C \).
To make this solution more explicit, exponentiate both sides, transforming the logarithmic equation into \( |\sec(y) + \tan(y)| = C e^{\frac{x^2}{2}} \). This step removes the logarithm and presents the solution in a clear, expoential form.

• The general solution captures all solutions through the arbitrary constant \( C \).
• Expressing the solution revolves around functions of \( y \) in terms of an exponential \( x \).

Being able to express the solution in terms of familiar functions provides insight into the behavior of the system modeled by the differential equation, and speaks to the broad applicability of the solution under varying initial conditions.