Question

Find the general solution to the differential equation.\(y^{\prime}=\tan (y) x\)

Short answer

\(\sin(y) = C' e^{\frac{x^2}{2}}\).

Step-by-step solution

Separate Variables

Begin by separating variables in the given differential equation \( y' = \tan(y) x \). This involves rearranging the terms so that each side of the equation depends only on one variable. We have \( \frac{dy}{\tan(y)} = x \, dx \).

Integrate Both Sides

Integrate both sides of the equation. The left side becomes \( \int \frac{1}{\tan(y)} \, dy = \int \cot(y) \, dy \), and the right side becomes \( \int x \, dx \). Integrating these gives \( \ln |\sin(y)| = \frac{x^2}{2} + C \), where \( C \) is the constant of integration.

Solve for the General Solution

Exponentiate both sides to solve for \( y \). We have \( |\sin(y)| = e^{\frac{x^2}{2} + C} \), which can be rewritten as \( \sin(y) = \pm e^{\frac{x^2}{2} + C} \). To simplify, let \( C' = e^C \), hence \( \sin(y) = C' e^{\frac{x^2}{2}} \). This is the general solution.

Key concepts

Separation of Variables

Separation of variables is a crucial technique in solving differential equations, especially when both sides can be expressed using only one variable each. It allows us to easily integrate and find a solution. By identifying an equation like \( y' = \tan(y) x \), you can rearrange the terms to separate the variables. This means isolating \( y \)-related terms on one side and \( x \)-related terms on the other. In our example, we rewrite it as \( \frac{dy}{\tan(y)} = x \, dx \),
where \( \frac{dy}{\tan(y)} \) depends only on \( y \), and \( x \, dx \) depends only on \( x \). Achieving this separation is key.

Once variables are separated, each side can be integrated independently, a method that often simplifies finding the solution of the equation.

Integration

Integration plays a pivotal role in solving differential equations after variable separation. It involves computing the antiderivatives of both sides of the equation separately.

For the left-hand side, \( \int \frac{1}{\tan(y)} \, dy \) can be expressed as \( \int \cot(y) \, dy \). Integrating \( \cot(y) \) yields \( \ln |\sin(y)| \).
For the right-hand side, \( \int x \, dx \) leads to the result \( \frac{x^2}{2} \).

This step is crucial because integrating transforms the differential equation into an equation without derivatives. Remember to include a constant of integration, \( C \), which represents a family's solutions rather than just one. Always check that integration is done correctly, as even small errors can lead to incorrect results.

General Solution

The general solution of a differential equation represents a family of functions that includes every possible solution based on the initial condition. After integration, we must express the equation in terms of \( y \). In this example, solving \( \ln |\sin(y)| = \frac{x^2}{2} + C \),
we exponentiate both sides to eliminate the logarithm:
\[ |\sin(y)| = e^{\frac{x^2}{2} + C} \]
This can be further simplified by letting \( C' = e^C \), resulting in \( \sin(y) = \pm C' e^{\frac{x^2}{2}} \).

This general solution describes all potential functions \( y(x) \) that satisfy the original differential equation. It's considered 'general' because it incorporates constant \( C' \), captured from initial value conditions.
Fixing \( C' \) leads to a particular solution, narrowing down to a specific function from the general family.