Problem 107
Question
The differential equation whose solution is \(A x^{2}+B y^{2}\) \(=1\), where \(A\) and \(B\) are arbitrary constants is of [2006] (A) second order and second degree (B) first order and second degree (C) first order and first degree (D) second order and first degree
Step-by-Step Solution
Verified Answer
First order and first degree.
1Step 1: Understand the Given Solution
The solution to the differential equation is given as \( A x^2 + B y^2 = 1 \). This is an implicit equation representing a relation between \( x \) and \( y \). Constants \( A \) and \( B \) are arbitrary.
2Step 2: Differentiate the Given Solution
Differentiate \( A x^2 + B y^2 = 1 \) with respect to \( x \). Using the chain rule, this gives \( 2Ax + 2By \frac{dy}{dx} = 0 \).
3Step 3: Rearrange the Differential Equation
Rearrange the differentiated equation to express \( \frac{dy}{dx} \) explicitly: \[ By \frac{dy}{dx} = -Ax \] This implies:\[ \frac{dy}{dx} = -\frac{Ax}{By} \]
4Step 4: Determine the Order and Degree
The first derivative \( \frac{dy}{dx} \) appears and no higher derivatives, indicating the differential equation is first order. The equation \( By \frac{dy}{dx} = -Ax \) is already solved for \( \frac{dy}{dx} \) without needing to take powers, indicating it is first degree.
Key Concepts
Understanding Implicit DifferentiationOrder of Differential EquationsDegree of Differential Equations
Understanding Implicit Differentiation
Implicit differentiation is a method used to differentiate equations that define a relationship between variables, indirectly or implicitly, rather than explicitly defining one variable in terms of another. For instance, the equation \( A x^2 + B y^2 = 1 \) describes the relationship between \( x \) and \( y \) without explicitly solving for \( y \) as a function of \( x \). This is where implicit differentiation comes into play.
Typically, differentiation involves taking the derivative of both sides of the equation with respect to the independent variable, often \( x \). When you differentiate the given equation, you apply the chain rule, especially on parts where the independent variable is intermixed with the dependent variable.
In our example, differentiating \( A x^2 + B y^2 = 1 \) with respect to \( x \) requires us to apply the chain rule, resulting in the derivative \( 2Ax + 2By \frac{dy}{dx} = 0 \).
From this result, we solve for \( \frac{dy}{dx} \), which provides the slope of the tangent line to the curve defined by the implicit equation at any point \((x,y)\). This process is crucial in implicitly defined equations, as it helps us understand how the dependent variable \( y \) behaves as \( x \) changes.
Typically, differentiation involves taking the derivative of both sides of the equation with respect to the independent variable, often \( x \). When you differentiate the given equation, you apply the chain rule, especially on parts where the independent variable is intermixed with the dependent variable.
In our example, differentiating \( A x^2 + B y^2 = 1 \) with respect to \( x \) requires us to apply the chain rule, resulting in the derivative \( 2Ax + 2By \frac{dy}{dx} = 0 \).
From this result, we solve for \( \frac{dy}{dx} \), which provides the slope of the tangent line to the curve defined by the implicit equation at any point \((x,y)\). This process is crucial in implicitly defined equations, as it helps us understand how the dependent variable \( y \) behaves as \( x \) changes.
Order of Differential Equations
The order of a differential equation is determined by the highest derivative that appears in the equation. In other words, the order tells us how many times the function is being differentiated. This is key to understanding the complexity and behavior of the solution to the differential equation.
In our example, after differentiating, we get the equation \( By \frac{dy}{dx} = -Ax \). Here, \( \frac{dy}{dx} \) is the first derivative and no higher derivatives appear, signifying that this is a first-order differential equation.
First-order differential equations are often simpler to solve and they describe a wide range of phenomena such as exponential growth and decay, or simple harmonic motion. They provide insights into the problem without unnecessary complexity, making them quite prevalent in initial studies of calculus and differential equations.
In our example, after differentiating, we get the equation \( By \frac{dy}{dx} = -Ax \). Here, \( \frac{dy}{dx} \) is the first derivative and no higher derivatives appear, signifying that this is a first-order differential equation.
First-order differential equations are often simpler to solve and they describe a wide range of phenomena such as exponential growth and decay, or simple harmonic motion. They provide insights into the problem without unnecessary complexity, making them quite prevalent in initial studies of calculus and differential equations.
Degree of Differential Equations
The degree of a differential equation is the power of the highest-order derivative in the equation when it is in the standard polynomial form. Unlike order, degree does not consider the number of derivatives but their powers.
For the equation we derived, \( By \frac{dy}{dx} = -Ax \), the first derivative \( \frac{dy}{dx} \) appears to the power of one. Therefore, the differential equation is of the first degree.
Degree is an important consideration, especially when dealing with nonlinear differential equations. It determines the type and complexity of the solution techniques that can be applied. While higher degree differential equations tend to be more complex and require more sophisticated methods or approximations, a first-degree differential equation like this one can often be solved through direct techniques such as separation of variables or integrating factors.
For the equation we derived, \( By \frac{dy}{dx} = -Ax \), the first derivative \( \frac{dy}{dx} \) appears to the power of one. Therefore, the differential equation is of the first degree.
Degree is an important consideration, especially when dealing with nonlinear differential equations. It determines the type and complexity of the solution techniques that can be applied. While higher degree differential equations tend to be more complex and require more sophisticated methods or approximations, a first-degree differential equation like this one can often be solved through direct techniques such as separation of variables or integrating factors.
Other exercises in this chapter
Problem 105
The differential equation representing the family of curves \(y^{2}=2 c(x+\sqrt{c})\) where \(c>0\), is a parameter, is of order and degree as follows: \([2005]
View solution Problem 106
If \(x \frac{d y}{d x}=y(\log y-\log x+1)\), then the solution of the equation is (A) \(y \log \left(\frac{x}{y}\right)=c x\) (B) \(x \log \left(\frac{y}{x}\rig
View solution Problem 108
The differential equation of all circles passing through the origin and having their centres on the \(x\)-axis is (A) \(x^{2}=y^{2}+x y \frac{d y}{d x}\) [2007]
View solution Problem 109
The solution of the differential equation \(\frac{d y}{d x}=\frac{x+y}{x}\) satisfying the condition \(y(1)=1\) is \([2008]\) (A) \(y=\ln x+x\) (B) \(y=x \ln x+
View solution