Problem 10
Question
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: (a) order 1, degree 2 (b) order 1 , degree 1 (c) order 1, degree 3 (d) order 2, degree 2
Step-by-Step Solution
Verified Answer
The differential equation is of order 1, degree 1 (option b).
1Step 1: Write the given equation
The equation of the family of curves is given as \( y^2 = 2c(x+\sqrt{c}) \). We know that \( c \) is a positive parameter.
2Step 2: Differentiate with respect to x
To eliminate the parameter \( c \), differentiate the given equation with respect to \( x \):\[ \frac{d}{dx}(y^2) = \frac{d}{dx}[2c(x+\sqrt{c})] \]This results in:\[ 2y \frac{dy}{dx} = 2c \]
3Step 3: Simplify and solve for c
Reorganize the differentiated equation to solve for \( c \):\[ c = y \frac{dy}{dx} \]
4Step 4: Substitute c back into the original equation
Substitute \( c = y \frac{dy}{dx} \) back into the original equation to eliminate \( c \):\[ y^2 = 2(y \frac{dy}{dx})(x+\sqrt{y \frac{dy}{dx}}) \]
5Step 5: Determine degree and order
Since our last step only involves first derivatives \( \frac{dy}{dx} \) and no higher derivatives are present, the order of the resulting differential equation is 1. The degree is the highest power of the derivative present, here it is 1 for \( \frac{dy}{dx} \). Thus, the order is 1 and the degree is 1.
Key Concepts
Order of Differential EquationsDegree of Differential EquationsFamily of Curves
Order of Differential Equations
The order of a differential equation is defined by the highest derivative present in the equation. In simpler terms, it's the term with the most 'd's in its derivative notation form. Assessing the order of a differential equation is crucial because it informs us about the complexity and the types of methods that might be suitable for solving it.
When we worked through the example with the family of curves, the resulting differential equation only involved the first derivative, denoted as \(\frac{dy}{dx}\). No higher derivatives appeared like \(\frac{d^2y}{dx^2}\) or more. Thus, in this exercise, we concluded that the order is 1.
Remember:
When we worked through the example with the family of curves, the resulting differential equation only involved the first derivative, denoted as \(\frac{dy}{dx}\). No higher derivatives appeared like \(\frac{d^2y}{dx^2}\) or more. Thus, in this exercise, we concluded that the order is 1.
Remember:
- The order is determined by the highest derivative in the equation.
- It's a fundamental classification aspect of differential equations.
Degree of Differential Equations
The degree, on the other hand, is determined by the highest power of the highest order derivative present after the equation is made free from fractions and radicals involving derivatives. It's another level of classification that helps to further understand the behavior and solutions of differential equations.
In the exercise, after substituting and differentiating, we only dealt with the first derivative \(\frac{dy}{dx}\), and it appeared to the first power. Meaning, there were no squares or cubes of this derivative term involved, making the degree 1.
Key notes about the degree:
In the exercise, after substituting and differentiating, we only dealt with the first derivative \(\frac{dy}{dx}\), and it appeared to the first power. Meaning, there were no squares or cubes of this derivative term involved, making the degree 1.
Key notes about the degree:
- The degree must be a whole number.
- It helps in understanding the nature of the differential equation's solutions.
Family of Curves
A family of curves is essentially a set or collection of curves that are generally defined by an equation with parameters. The parameter can vary, leading to different specific members of the family, but all share a common structure with respect to their form.
In this exercise, we started with an equation involving a parameter \(c\). This parameter describes a family, and each particular choice of \(c\) gives us a different curve. The differential equation we derived represents all possible curves in this family by eliminating this parameter.
Why we study families of curves:
In this exercise, we started with an equation involving a parameter \(c\). This parameter describes a family, and each particular choice of \(c\) gives us a different curve. The differential equation we derived represents all possible curves in this family by eliminating this parameter.
Why we study families of curves:
- They provide a comprehensive picture by uniting multiple solutions into one.
- It becomes easier to study complex systems by understanding the family behavior.
Other exercises in this chapter
Problem 8
The differential equation of all circles passing through the origin and having their centres on the x-axis is (a) \(y^{2}=x^{2}+2 x y \frac{d y}{d x}\) (b) \(y^
View solution Problem 9
The differential equation whose solution is \(A x^{2}+B y^{2}=1\) where \(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constants is of (a) second order and sec
View solution Problem 11
The differential equation for the family of circle \(x^{2}+y^{2}-2 a y=0\), where a is an arbitrary constant is (a) \(\left(x^{2}+y^{2}\right) y^{\prime}=2 x y\
View solution Problem 12
The degree and order of the differential equation of the family of all parabolas whose axis is \(x\) - axis, are respectively. (a) 2,3 (b) 2,1 (c) 1,2 (d) 3,2 .
View solution