Question

Integrate \(d P / d \theta-l \cot \theta P=0\) to find \(P(\theta)\).

Short answer

The solution to the given differential equation is \(P(\theta) = A (\sin\theta)^l\).

Step-by-step solution

Express Equations in Standard Form

First, express the equation in the standard form of first order linear differential equation. It can be expressed as \(d P / d \theta - l \cot \theta P = 0\). The ultimate goal is to solve this equation to find \(P (\theta)\).

Separate the Variables

The next step is to separation of variables. Arranging the terms, we can write this equation as \(d P / P = l \cot \theta d \theta\). Here the variables are separated such that the terms involving P are on one side and the terms involving \(\theta\) are on the other side.

Integrate Both Sides

The separated equation is then integrated on both sides. This leads to \(\int d P / P = l \int \cot \theta d \theta\). The integral of 1/P with respect to P is \(ln|P|\) and the integral of \(\cot \theta\) with respect to \(\theta\) is \(ln|\sin\theta|\). Hence, the solution to the above equation becomes \( ln|P(\theta)| = l ln|\sin\theta| + C\), where C is the constant of integration.

Simplify Equation

The final step is to clear out the natural log from the equation. This is done by exponentiating each side, which gives: \(|P(\theta)| = |\sin\theta|^l e^C\). Now we can simply express \(P(\theta)\) as \(P(\theta) = A \sin\theta)^l\), where A is \(e^C\) (a constant).

Key concepts

Separation of Variables

Separation of variables is a key technique used to solve first order linear differential equations. This method involves rearranging an equation so that all terms involving one variable are on one side of the equation, while all terms involving another variable are on the opposite side.
For instance, in the example of integrating \(dP/d\theta - l \cot\theta P = 0\), you first express it to isolate terms involving \(P\) and \(\theta\). This gives \(dP/P = l \cot\theta d\theta\).
This separation allows each side of the equation to be integrated independently, making the next step, which is applying integration techniques, more straightforward. Once the variables are separated, we can apply the appropriate integration techniques to solve the equation. This method is ideal for equations where the variables are explicitly separable, simplifying the process of finding the solution significantly.

Integration Techniques

Once we have successfully separated the variables, we need to integrate both sides of the equation. The goal is to find an antiderivative that represents each side of the equation.
In our example, we have \({\int \frac{dP}{P} = \int l \cot\theta d\theta}\). The left side, which is \(\int \frac{dP}{P}\), can be integrated directly to give \(\ln|P|\). The right side involves integrating the trigonometric function \(\cot\theta\), which results in \(\ln|\sin\theta|\).
It is crucial to be comfortable with basic integration techniques, such as substitutions and knowledge of common functions, to manage equations like this. These skills are essential for resolving more complex differential equations, building a foundation for further mathematical studies.

Constants of Integration

After integrating, the next step is to include the constant of integration, which arises because indefinite integrals are defined up to a constant. This constant reflects the fact that there are infinitely many antiderivatives each differing by a constant.
In our solution, the inclusion of a constant after integrating gives the equation \(\ln|P(\theta)| = l \ln|\sin\theta| + C\), where \(C\) is the integration constant. When we solve for \(P(\theta)\), we actually express this as \(P(\theta) = A(\sin\theta)^l\), where \(A = e^C\).
The constant of integration plays a vital role in the completeness of the solution, ensuring that all possible solutions are considered. Understanding and incorporating this constant correctly is essential to solving first order linear differential equations accurately.