Question

Find the particular solution to the differential equation \(\frac{d u}{d t}=\tan u\) that passes through \(\left(1, \frac{\pi}{2}\right)\), given that \(u=\sin ^{-1}\left(e^{C+t}\right)\) is a general solution.

Short answer

The particular solution is \( u = \sin^{-1}(e^{-1+t}) \).

Step-by-step solution

Understand the Problem

We need to find the particular solution to the differential equation \( \frac{d u}{d t} = \tan u \) that passes through the point \((1, \frac{\pi}{2})\). The general solution is given as \( u = \sin^{-1}(e^{C+t}) \). This means we need to determine the constant \( C \) that makes the solution satisfy the initial condition.

Substitute the Point into the General Solution

Substitute \( t = 1 \) and \( u = \frac{\pi}{2} \) into the general solution. This gives us the equation \( \frac{\pi}{2} = \sin^{-1}(e^{C+1}) \).

Solve for the Exponential Expression

Since \( \sin^{-1}(x) = \frac{\pi}{2} \) when \( x = 1 \), we set \( e^{C+1} = 1 \). This is because \( \sin\left(\frac{\pi}{2}\right) = 1 \).

Solve for the Constant C

Take the natural logarithm of both sides: \( \ln(e^{C+1}) = \ln(1) \). This simplifies to \( C + 1 = 0 \), leading to the solution \( C = -1 \).

Write the Particular Solution

With \( C = -1 \), substitute back into the general solution. The particular solution is \( u = \sin^{-1}(e^{-1+t}) \).

Key concepts

Particular Solutions

When dealing with differential equations, a particular solution refers to a solution whose constants have been specifically evaluated so that it satisfies given conditions, usually initial or boundary conditions. In the provided exercise, the goal is to find a particular solution to the differential equation \(\frac{du}{dt} = \tan u\), which passes through a specific point \((1, \frac{\pi}{2})\).

To find the particular solution, we start with the general solution given by \(u = \sin^{-1}(e^{C+t})\). By substituting the initial condition into the general solution, we calculate the constant \(C\), allowing us to form the particular solution. Therefore, through this process, we tailor the general solution to satisfy our exact conditions, which often leads to just one unique answer. The result of this problem is \(u = \sin^{-1}(e^{-1+t})\), a particular solution that precisely matches the requirements given.

General Solutions

A general solution to a differential equation is a solution that encompasses all possible solutions within a particular family of solutions. It represents the general form of the solution containing one or more arbitrary constants, typically representing a family of curves. In our case, the general solution to the differential equation \(\frac{du}{dt} = \tan u\) is \(u = \sin^{-1}(e^{C+t})\).

The constant \(C\) in this equation is crucial, as it allows for the generation of more specific solutions—particular solutions—when specific initial or boundary conditions are applied. This form of solution affords flexibility because changing \(C\) can shift or alter the specific curve of the solution. It symbolizes all possible scenarios that the solution can take on before locking it into a specific condition or data point. In essence, the general solution serves as the starting point before imposing additional conditions to determine unique particular solutions.

Initial Conditions

Initial conditions are vital pieces of information used to find a specific (or particular) solution to a differential equation. They help pinpoint which solution in the family of solutions—represented by the general solution—is the one that matches a real-world scenario or specific problem setting.

For instance, in our exercise, the initial condition given is the point \((1, \frac{\pi}{2})\). By substituting \(t = 1\) and \(u = \frac{\pi}{2}\) into the general solution \(u = \sin^{-1}(e^{C+t})\), we determine the exact value of the constant \(C\). This transforms our general solution into a particular one, revealing the precise path the solution takes that not only satisfies the differential equation but also adheres to the initial condition. Initial conditions, in essence, dictate which unique path or curve out of a whole family is relevant to a specific problem. This is akin to choosing one specific road out of many possible ways to reach a destination based on clear, predefined instructions or requirements.