Question

Solve the following differential equations. \(y y^{\prime}-2 y^{2} \cot x=\sin x \cos x\)

Short answer

The general solution is: \( y^{2} = 2 y \ln|\sin x| + \sin x \cos x + c \).

Step-by-step solution

Rewrite the Equation

The given differential equation is \( y y^{\prime}-2 y^{2} \cot x=\sin x \cos x \). To solve this, we first rewrite it in a more workable form. Let's isolate the derivative term: \( y y^{\prime} = 2 y^{2} \cot x + \sin x \cos x \).

Simplify the Equation

Divide the entire equation by \( y \): \( y^{\prime} = 2 y \cot x + \frac{\sin x \cos x}{y} \).

Express as a Linear Differential Equation

We can rearrange the terms to: \( y^{\prime} - 2 y \cot x = \frac{\sin x \cos x}{y} \). Notice that this can be solved by separation of variables.

Separate Variables

Separate the variables: \( y \frac{dy}{dx} = 2 y \cot x + \sin x \cos x \).Divide by \( y \): \( \frac{dy}{dx} = 2 \cot x + \frac{\sin x \cos x }{y} \).Separate to isolate \( y \): \( y \frac{ dy}{dx} = 2 \cot x \cdot y + \sin x \cos x \cdot \frac{1}{y} \).

Integrate Both Sides

Integrate on both sides: Integrate the left side with respect to \( y \) and the right side with respect to \( x \): \(\int y dy = \int 2 \cot x \cdot y dx + \int \frac{\sin x \cos x}{y} dx\).

Solve the Integrals

Integrate each term separately: \( \frac{1}{2} y^{2} = 2 y \ln|\sin x| + \frac{1}{2 y} \sin 2x + C \).

Simplify the Solution

Now simplify the equation and solve for y, the simplified general solution is: \( y^{2} = 2 y \ln|\sin x| + \sin x \cos x + c \). Rearrange terms to solve for \( y \).

Key concepts

Separation of Variables

One common method for solving differential equations is the separation of variables technique. This method works best for equations that can be rearranged such that one side includes all terms involving the dependent variable and its derivatives, and the other side includes all terms with the independent variable.
In the given problem, the differential equation is initially written as:
\[ yy' - 2y^2 \text{cot}(x) = \text{sin}(x) \text{cos}(x) \]
To use separation of variables, we rearranged terms to isolate the derivative term, resulting in:
\[ y' = 2y \text{cot}(x) + \frac{\text{sin}(x) \text{cos}(x)}{y} \]
By separating the variables in the above equation, we express it in a form where the variables are isolated on each side:
\[ y \frac{dy}{dx} = 2 \text{cot}(x) y + \frac{\text{sin}(x) \text{cos}(x)}{y} \]
Dividing through by y and further rearranging results in separated variables:
\[ \frac{dy}{dx} = 2 \text{cot}(x) + \frac{\text{sin}(x) \text{cos}(x)}{y} \]
This allows us to integrate each side separately to find the solution.
Key steps in separation of variables include:

• Isolating the variables on opposite sides of the equation.
• Ensuring the original differential equation is manipulable into separable form.
• Careful integration of both sides after separation.

Linear Differential Equations

Linear differential equations are a specific type that can generally be written in the form:
\[ y' + P(x)y = Q(x) \]
These equations are called 'linear' because they involve no powers or products of the dependent variable and its derivatives higher than one. In our case, we start with:
\[ yy' - 2y^2 \text{cot}(x) = \text{sin}(x)\text{cos}(x) \]
After isolating the derivative term and rearranging, we get:
\[ y' - 2y \text{cot}(x) = \frac{\text{sin}(x)\text{cos}(x)}{y} \]
Notice this form is similar to the linear standard form:
\[ y' + P(x)y = Q(x) \], where:
\[ P(x) = -2\text{cot}(x) \] and \[ Q(x) = \frac{\text{sin}(x)\text{cos}(x)}{y} \].
Understanding the form and traits of linear differential equations helps to systematically solve them using methods like integrating factors and direct integration.
Key traits of linear differential equations:

• Involves no products or powers of y and its derivatives.
• Can often be solved using specific techniques tailored for linear equations like integrating factors.
• General solutions often involve integrals and constants of integration that need to be solved explicitly.

Integration Techniques

Solving differential equations often involves integration, as it is the process used to find the antiderivative of functions. In our solution:
Integrate each term separately:
\[ \frac{1}{2} y^2 = 2y \text{ln}|\text{sin}(x)| + \frac{1}{2y} \text{sin}(2x) + C \]
Integration techniques used in solving differential equations include:

• Direct integration when the integrand is straightforward.
• Using trigonometric identities to simplify complex integrands such as \[ \text{sin}(2x) \].
• Substitution technique, when variables change to simplify the integral form.

In our specific example, after dividing and arranging the terms to use the separation of variables, solving each side involves integrating:
\[ \frac{1}{2} y^2 = \text{Integral of} \big( 2y \text{ln}|\text{sin}(x)| + \frac{\text{sin}(2x)}{2y} \big) + C \]
Simplifying the integrals by recognizing and using identities such as:
\[ \text{sin}(2x) = 2\text{sin}(x)\text{cos}(x) \] and employing partial fraction decomposition when necessary, are key steps to integrate effectively.