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Chapter 1: Problem 3

Form the differential equation in each of the following cases by eliminating the parameters mentioned against each. $$ y=m x+\frac{a}{m} $$

Short Answer

Expert verified
The differential equation is \( y \frac{dy}{dx} = x \left( \frac{dy}{dx} \right)^2 + a \).

Step by step solution

01

Understand the Given Equation

The given equation is \( y = mx + \frac{a}{m} \) where \( m \) is the parameter that needs to be eliminated. The goal is to form a differential equation that does not include the parameter \( m \).
02

Differentiate with Respect to x

Differentiate the equation \( y = mx + \frac{a}{m} \) with respect to \( x \). This gives \( \frac{dy}{dx} = m \), which provides a relationship between \( y \), \( x \), and \( m \).
03

Solve for m

Using the differentiated equation \( \frac{dy}{dx} = m \), solve for the parameter \( m \): \( m = \frac{dy}{dx} \).
04

Substitute m Back into the Original Equation

Substitute \( m = \frac{dy}{dx} \) back into the original equation \( y = mx + \frac{a}{m} \). This results in the equation \( y = \frac{dy}{dx} \cdot x + \frac{a}{\frac{dy}{dx}} \).
05

Simplify the Equation

Simplify the equation from Step 4. Multiply throughout by \( \frac{dy}{dx} \) to eliminate the fraction: \( y \frac{dy}{dx} = \frac{dy}{dx} \cdot x \cdot \frac{dy}{dx} + a \).
06

Rearrange and Generalize

The final form of the differential equation is: \( y \frac{dy}{dx} = x \left( \frac{dy}{dx} \right)^2 + a \). This equation is free from the parameter \( m \) and describes the relationship given in the original problem.

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Key Concepts

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

Parameter Elimination
Parameter elimination is a crucial step in forming differential equations. In the context of the given problem, we started with an equation involving the parameter \( m \):
  • All parameters need to be removed to derive a pure differential equation.
  • The parameter \( m \) is initially part of the equation \( y = mx + \frac{a}{m} \), and the goal is to eliminate it entirely.
To eliminate \( m \), we utilize differentiation and substitution. This involves differentiating the original equation to express \( m \) in terms of derivatives, allowing us to rewrite the equation without \( m.\) By substituting the expression for \( m \) back into the equation, we can transform it into a differential equation that only contains variables and their derivatives.
Differentiation
Differentiation is a fundamental tool used extensively in calculus to derive relationships between changing quantities. In this problem:
  • We started by differentiating the given equation \( y = mx + \frac{a}{m} \) with respect to \( x \).
  • This operation yielded \( \frac{dy}{dx} = m \), a simple derivative that points out how \( y \) changes as \( x \) changes under this linear equation.
Differentiation allows us to isolate the parameter \( m \), giving us an expression \( m = \frac{dy}{dx} \). This step is critical as it connects the dependent and independent variables via their rate of change, which is the derivative.
Formation of Differential Equations
Forming a differential equation is all about expressing a relationship without parameters.
  • The original equation and its differentiated form allowed us to eliminate \( m \) by expressing it as a derivative: \( m = \frac{dy}{dx} \).
  • By substituting \( m \) back into the original equation, we reshaped it in terms of the derivative and other existing variables.
The end goal is to express the relationship in the absence of any parameters. In our case, this led to the equation: \[ y \frac{dy}{dx} = x \left( \frac{dy}{dx} \right)^2 + a \] This new equation is a differential equation since it contains the derivatives \( \frac{dy}{dx} \) free from the parameter \( m \). This formula describes how changes in \( x \) and \( y \) are interrelated.
Mathematical Problem Solving
Mathematical problem-solving skills are essential for tackling differential equations effectively. When faced with a challenge like the one in this exercise:
  • First, clearly understand the problem and identify the given quantities.
  • Plan a strategy that will lead to eliminating parameters.
  • Differentiation acts as a potent strategy to derive relations between variables.
The step-by-step solution maps out a logical approach, starting with simplifying what we have, differentiating to discover relations, solving for parameters, and finally substituting back to form a parameter-free equation. This approach not only aids in solving similar problems but also reinforces core calculus concepts, emphasizing critical thinking and methodical work. Successful problem-solving is about patience and an attentive approach to mathematical processes.

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Most popular questions from this chapter

Show that the differential equation obtained from \(y=A e^{a x} \cos b x+B e^{a x} \sin b x\) where \(A\) and \(B\) are arbitrary constants is \(y^{\prime \prime}-2 a y^{\prime}+\left(a^{2}+b^{2}\right) y=0\).

Form the differential equation for the family of circles, touching the \(x\) -axis at \((0,0)\). (Hint: \(x^{2}+y^{2}-2 f y=0, f\) parameter)

Form the differential equation in each of the following cases by eliminating the parameters mentioned against each. $$ y=a x+b x^{2} $$

Form the differential equation in each of the following cases by eliminating the parameters mentioned against each. $$ (x-h)^{2}+(y-k)^{2}=r^{2} $$

Find the differential equation by eliminating \(c\) from \(y=c x+x^{3}\).
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