Problem 13
Question
The order and degree of the differential equation \(\left(1+3 \frac{d y}{d x}\right)^{2 / 3}=4 \frac{d^{3} y}{d x^{3}}\) are \(\quad[\mathbf{2 0 0 2}]\) (a) \(\left(1, \frac{2}{3}\right)\) (b) \((3,1)\) (c) \((3,3)\) (d) \((1,2)\)
Step-by-Step Solution
Verified Answer
The order is 3 and the degree is 1. So the answer is (b) (3,1).
1Step 1: Identify the Derivative Orders
Examine the differential equation \( \left(1+3 \frac{d y}{d x}\right)^{2/3}=4 \frac{d^{3} y}{d x^{3}} \). Notice the derivatives \( \frac{d y}{d x} \) and \( \frac{d^3 y}{d x^3} \). This indicates that the highest derivative present is the third derivative, so the order of the differential equation is 3.
2Step 2: Determine the Degree of the Differential Equation
The degree of a differential equation is defined as the power of the highest order derivative, after the equation is made polynomial in terms of the derivatives. In \( \left(1+3 \frac{d y}{d x}\right)^{2/3}=4 \frac{d^{3} y}{d x^{3}} \), \( \frac{d^3 y}{d x^3} \) is raised to the power of 1. Even though \( \left(1+3 \frac{d y}{d x}\right)^{2/3} \) has a fractional power, it does not involve the highest derivative and thus does not affect the degree.
3Step 3: Finalize Order and Degree
Considering that the highest derivative is \( \frac{d^3 y}{d x^3} \) with a degree of 1, the order of the differential equation is 3 and the degree is also 1.
Key Concepts
Order of differential equationsDegree of differential equationsThird order derivative
Order of differential equations
In differential equations, the order is defined by the highest derivative present in the equation. To identify the order, one must examine all the derivatives found in the equation.
For example, consider the equation \( \left(1+3 \frac{d y}{d x}\right)^{2/3}=4 \frac{d^{3} y}{d x^{3}} \). Here, the derivatives present are \( \frac{d y}{d x} \) and \( \frac{d^3 y}{d x^3} \).
As the third derivative \( \frac{d^3 y}{d x^3} \) is the one with the highest order in this equation, it determines the order of the differential equation. Hence, the order is 3.
For example, consider the equation \( \left(1+3 \frac{d y}{d x}\right)^{2/3}=4 \frac{d^{3} y}{d x^{3}} \). Here, the derivatives present are \( \frac{d y}{d x} \) and \( \frac{d^3 y}{d x^3} \).
As the third derivative \( \frac{d^3 y}{d x^3} \) is the one with the highest order in this equation, it determines the order of the differential equation. Hence, the order is 3.
- Highlight: Always locate the highest derivative to determine the order.
- It's crucial for identifying the behavior of solutions.
Degree of differential equations
The degree of a differential equation is linked to the power of the highest order derivative. However, the equation must first be expressed in polynomial form with respect to its derivatives.
In our original equation, \( \left(1+3 \frac{d y}{d x}\right)^{2/3}=4 \frac{d^{3} y}{d x^{3}} \), the focus is on the highest order derivative, which is \( \frac{d^3 y}{d x^3} \). This derivative appears to the first power.
Even though the term \( \left(1+3 \frac{d y}{d x}\right)^{2/3} \) is a fractional power, it does not contain the highest order derivative. Consequently, it doesn't affect the degree of the equation. Thus, the degree is 1.
In our original equation, \( \left(1+3 \frac{d y}{d x}\right)^{2/3}=4 \frac{d^{3} y}{d x^{3}} \), the focus is on the highest order derivative, which is \( \frac{d^3 y}{d x^3} \). This derivative appears to the first power.
Even though the term \( \left(1+3 \frac{d y}{d x}\right)^{2/3} \) is a fractional power, it does not contain the highest order derivative. Consequently, it doesn't affect the degree of the equation. Thus, the degree is 1.
- Important Note: Only the power of the highest order derivative counts for determining degree.
- Fractional powers in other terms do not impact degree.
Third order derivative
The third order derivative, denoted as \( \frac{d^3 y}{d x^3} \), corresponds to the rate of change of the rate of change of the rate of change of the function \( y \).
In other words, while the first derivative \( \frac{d y}{d x} \) gives the rate of change of \( y \), which is often interpreted as its slope or velocity, the second derivative \( \frac{d^2 y}{d x^2} \) represents how that rate of change is itself changing, typically regarded as acceleration.
The third order derivative then delves deeper, offering insights into how this acceleration varies.
In other words, while the first derivative \( \frac{d y}{d x} \) gives the rate of change of \( y \), which is often interpreted as its slope or velocity, the second derivative \( \frac{d^2 y}{d x^2} \) represents how that rate of change is itself changing, typically regarded as acceleration.
The third order derivative then delves deeper, offering insights into how this acceleration varies.
- This can be crucial in more complex systems where understanding the subtle shifts in dynamics is essential.
- It provides higher-level insights into behaviors such as jerk in motion.
Other exercises in this chapter
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 Problem 15
If \(y=\left(\frac{2}{\pi} x-1\right) \operatorname{cosec} x\) is the solution of the differential equation, \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\mathrm{p}(x)
View solution Problem 16
If \(y=y(x)\) is the solution of the differential equation \(\frac{5+\mathrm{e}^{x}}{2+y} \cdot \frac{\mathrm{d} y}{\mathrm{~d} x}+\mathrm{e}^{x}=0\) satisfying
View solution