The order and degree of the differential equation \(\rm k \dfrac{dy}{dx}=\displaystyle\int \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}dx\) are respectively
2 and 3
Understanding Order and Degree of a Differential Equation
The question asks for the order and degree of the given differential equation: \(\rm k \dfrac{dy}{dx}=\displaystyle\int \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}dx\).
Let's break down how to find the order and degree of this differential equation.
What is the Order?
The order of a differential equation is the order of the highest derivative present in the equation.
What is the Degree?
The degree of a differential equation is the highest power of the highest order derivative present in the equation, after it has been made free from radicals and fractions concerning the derivatives.
Step-by-Step Solution
We are given the differential equation:
\(\rm k \dfrac{dy}{dx}=\displaystyle\int \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}dx\)
This equation involves an integral. To find the order and degree, the equation must be free from integral signs. We can eliminate the integral by differentiating both sides with respect to \(x\).
Differentiating both sides with respect to \(x\):
\(\dfrac{d}{dx}\left(\rm k \dfrac{dy}{dx}\right) = \dfrac{d}{dx}\left(\displaystyle\int \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}dx\right)\)
Applying the rules of differentiation and the Fundamental Theorem of Calculus on the right side:
\(\rm k \dfrac{d^2y}{dx^2} = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}\)
Determining the Order
Now, examine the modified differential equation:
\(\rm k \dfrac{d^2y}{dx^2} = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}\)
The derivatives present are \(\dfrac{dy}{dx}\) (first order) and \(\dfrac{d^2y}{dx^2}\) (second order).
The highest order derivative present is \(\dfrac{d^2y}{dx^2}\).
Therefore, the order of the differential equation is 2.
Determining the Degree
The degree is the highest power of the highest order derivative (\(\dfrac{d^2y}{dx^2}\)), after the equation is made free from fractional or radical powers involving derivatives.
Our equation is currently:
\(\rm k \dfrac{d^2y}{dx^2} = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}\)
This equation has a fractional power \(\left(\frac{2}{3}\right)\) involving derivatives. To eliminate this fractional power, we need to raise both sides of the equation to the power of 3:
\(\left(\rm k \dfrac{d^2y}{dx^2}\right)^3 = \left(\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}\right)^3\)
Simplifying both sides:
\(\rm k^3 \left(\dfrac{d^2y}{dx^2}\right)^3 = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^2\)
Now, expand the right side:
\(\rm k^3 \left(\dfrac{d^2y}{dx^2}\right)^3 = 1^2 + 2\left(\dfrac{dy}{dx}\right)^2 + \left(\left(\dfrac{dy}{dx}\right)^2\right)^2\)
\(\rm k^3 \left(\dfrac{d^2y}{dx^2}\right)^3 = 1 + 2\left(\dfrac{dy}{dx}\right)^2 + \left(\dfrac{dy}{dx}\right)^4\)
The equation is now a polynomial in terms of the derivatives. The highest order derivative is \(\dfrac{d^2y}{dx^2}\).
The highest power of \(\dfrac{d^2y}{dx^2}\) in this equation is 3.
Therefore, the degree of the differential equation is 3.
So, the order and degree of the given differential equation are 2 and 3, respectively.
| Property | Value |
|---|---|
| Original Equation | \(\rm k \dfrac{dy}{dx}=\displaystyle\int \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}dx\) |
| After Differentiation | \(\rm k \dfrac{d^2y}{dx^2} = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}\) |
| After Removing Fractional Power | \(\rm k^3 \left(\dfrac{d^2y}{dx^2}\right)^3 = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^2\) |
| Highest Order Derivative | \(\dfrac{d^2y}{dx^2}\) |
| Order | 2 |
| Highest Power of Highest Order Derivative | 3 |
| Degree | 3 |
Revision Table: Order and Degree of Differential Equations
| Concept | Definition | Example |
|---|---|---|
| Order | Highest order of derivative present. | In \(\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^3 = 0\), the order is 2. |
| Degree | Highest power of the highest order derivative after making the equation a polynomial in derivatives. | In \(\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^3 = 0\), the highest order derivative is \(\dfrac{d^2y}{dx^2}\) with power 1. Degree is 1. |
| Degree (with fractional power) | Clear radicals/fractions on derivatives by raising to appropriate power. | In \(y' = \sqrt{1+(y'')^2}\), square both sides: \((y')^2 = 1+(y'')^2\). Highest order is \(y''\), power is 2. Degree is 2. |
Additional Information: Key Points on Order and Degree
- The order of a differential equation is always a positive integer.
- The degree of a differential equation is defined only if the equation can be expressed as a polynomial in derivatives. If the equation involves terms like \(e^{dy/dx}\) or \(\sin(d^2y/dx^2)\), the degree is undefined.
- To find the degree, always ensure the equation is rationalized and cleared of fractions with respect to derivatives.
- Integral signs in the equation must be removed by differentiation before determining the order and degree.
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