Question

Find the general solution of the differential equation and check the result by differentiation. $$ \frac{d r}{d \theta}=\pi $$

Short answer

The general solution of the differential equation is \(r = \pi\theta + C\). The validity of this solution is verified by its derivative.

Step-by-step solution

Solve the Integral

The given differential equation is \(\frac{d r}{d \theta} = \pi\). We can directly integrate both sides of it as no other dependent variable is involved. Upon integrating we get \(r = \pi\theta + C\), where C is the constant of integration.

Deriving the Solution

To verify the solution, we derive it with respect to \(\theta\). The derivative of the right side \(\pi\theta + C\) with respect to \(\theta\) is \(\pi\) which is exactly the same as given in the original differential equation.

Key concepts

Understanding Differential Equations

Let's start with the basics of differential equations. These are equations that involve derivatives, which measure how a quantity changes in response to changes in another quantity. In this case, we're looking at a simple form where the rate of change of a variable, here represented as \( \frac{d r}{d \theta} \), is constant. Think of \( \theta \) as time and \( r \) as the distance a car has traveled. If \( \frac{d r}{d \theta} = \pi \), it means the car moves at a speed of \( \pi \) units per unit of time, no matter what time it is.

The goal is to find the general solution which will tell us the value of \( r \) for any given \( \theta \) without imposing any specific initial condition. This solution will include all possible scenarios of starting points or initial conditions for \( r \) because of the constant of integration, which we'll discuss later. This solution is versatile and widely applicable, which is why it's often the first thing you'll want to find when dealing with a differential equation.

The Role of Integration in Solving Differential Equations

Integration is akin to detective work: you're reconstructing the 'whole' from its rate of change. When you integrate a function, you are essentially finding the quantity whose derivative is that function. In our exercise, integrating \( \frac{d r}{d \theta} = \pi \) relative to \( \theta \) gives us \( r = \pi\theta \), which represents the 'accumulated distance' in our car analogy as time \( \theta \) progresses.

When integrating, we always add a constant of integration, \( C \), which could be any real number. Why? Because there are infinitely many functions that have the same rate of changing, or derivative. This constant is vital as it encompasses all the possible starting points for our variable \( r \) at \( \theta = 0 \). It's a fundamental idea in calculus that reminds us of the family of potential solutions to our original problem.

Constant of Integration Explained

Imagine you’re given the task to trace your steps back after a hike without knowing the starting point. This is where the constant of integration, \( C \) in our solution \( r = \pi\theta + C \), comes into play. It represents our 'starting point' in the problem. In mathematical terms, any constant added during integration accounts for all the possible initial values that the original function, in this case, \( r \) could have taken.

Depending on the context, \( C \) can reflect initial conditions or integrate past information into the solution. If you know where your hike started, you can determine \( C \) precisely. But in the case of general solutions like ours, we leave \( C \) unspecified to acknowledge that the initial state might have been varied. This means \( r \) could have started from any value, and the expression \( r = \pi\theta + C \) is prepared to account for that variety, including every possible scenario in the general solution.