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Chapter 6: Problem 4

Find the general solution of the differential equation. \(\frac{d r}{d s}=0.05 s\)

Short Answer

Expert verified
The general solution of the differential equation \(\frac{d r}{d s}=0.05 s\) is \(r = 0.025 s^2 + c\), where c is a constant.

Step by step solution

01

Separate variables

Rewrite the differential equation \(\frac{d r}{d s}=0.05 s\) in separated variable form as \(dr = 0.05 s ds\). In this step, every s terms are moved to the right side of the equation and r terms to the left.
02

Integrate

Next, integrate both sides of the equation. We have \(\int dr = \int 0.05 s ds\). The integral of the left-hand side with respect to 'r' is just 'r'. On the right-hand side we have integral of '0.05s' with respect to 's' which is \(0.05 \int s ds = 0.05 [\frac{1}{2} s^2] = 0.025 s^2\).
03

Incorporate Integration Constant

We add a constant 'c' to the solution to account for constant of integration: \(r = 0.025 s^2 + c\).
04

Write Final Answer

Putting it all together, we obtain the general solution to the differential equation: \(r = 0.025 s^2 + c\), where c denotes an arbitrary constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Separation of Variables
When solving a differential equation, separation of variables is often the first step, especially for simple first-order equations. It involves rearranging the equation so that each variable appears on opposite sides of the equation. In our case, we begin with the equation \( \frac{d r}{d s} = 0.05 s \). To apply separation of variables:
  • Move all terms involving \( r \) to one side: \( dr \).
  • Move all terms involving \( s \) to the opposite side: \( 0.05s \, ds \).
This results in the separated form \( dr = 0.05 s \, ds \), meaning that we have effectively isolated each variable on its own side. This simplification allows us to proceed with integration, the next step in solving the differential equation.
Integration
Integration is the mathematical process of finding the integral of a function. In differential equations, once the variables have been separated, each side of the equation can be integrated separately. Let's do this with \( dr = 0.05 s \, ds \):
  • The left side, \( \int dr \), integrates to \( r \), because the integral of \( dr \) with respect to \( r \) is \( r \) itself.
  • For the right side, we need to find \( \int 0.05 s \, ds \).
  • This involves integrating \( 0.05s \) with respect to \( s \), which yields \( 0.025 s^2 \), calculated as follows: \( 0.05 \cdot \frac{1}{2} s^2 = 0.025 s^2 \).
By integrating both sides, we transform the differential equation into an algebraic equation: \[ r = 0.025 s^2 + c \] where \( c \) is the constant of integration, which we will discuss next.
Constants of Integration
The process of integration in calculus introduces a constant of integration because the derivative of a constant is zero. This means when integrating something, you cannot directly know what constant value may have been there before differentiation. In our equation, that constant is represented by \( c \): \[ r = 0.025 s^2 + c \]Here are a few important points to understand about constants of integration:
  • \( c \) is arbitrary and can represent any real number.
  • It accounts for any initial condition or initial value problem when solving real-world differential equations.
  • The presence of \( c \) means the solution represents a family of curves, each differing by the value of \( c \).
  • In practical problems, \( c \) can be determined if more information is given, such as an initial condition \((r_0, s_0)\).
Including the constant of integration is crucial for the general solution, ensuring that all possible solutions to the differential equation are represented.

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