What is the degree of the differential equation \(\frac{{{d}^{3}}y}{d{{x}^{3}}}+{{\left( \frac{dy}{dx} \right)}^{2}}-{{x}^{2}}\left( \frac{{{d}^{4}}y}{d{{x}^{4}}} \right)=0?\)
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Understanding Differential Equation Degree
The question asks for the degree of the given differential equation. To find the degree of a differential equation, we first need to determine its order. The order is the highest order of the derivative present in the equation. The degree is the power of the highest order derivative after the equation has been made free from radicals and fractions as far as derivatives are concerned. This definition applies only when the differential equation is a polynomial in its derivatives.
Analyzing the Given Differential Equation
The given differential equation is:
\(\frac{{{d}^{3}}y}{d{{x}^{3}}}+{{\left( \frac{dy}{dx} \right)}^{2}}-{{x}^{2}}\left( \frac{{{d}^{4}}y}{d{{x}^{4}}} \right)=0\)
Let's identify the derivatives present and their orders:
- \(\frac{dy}{dx}\) is a first-order derivative.
- \(\frac{{{d}^{3}}y}{d{{x}^{3}}}\) is a third-order derivative.
- \(\frac{{{d}^{4}}y}{d{{x}^{4}}}\) is a fourth-order derivative.
Determining the Order of the Differential Equation
The highest order derivative present in the equation is \(\frac{{{d}^{4}}y}{d{{x}^{4}}}\). Therefore, the order of this differential equation is 4.
| Derivative | Order |
|---|---|
| \(\frac{dy}{dx}\) | 1 |
| \(\frac{{{d}^{3}}y}{d{{x}^{3}}}\) | 3 |
| \(\frac{{{d}^{4}}y}{d{{x}^{4}}}\) | 4 |
Determining the Degree of the Differential Equation
Now, we look at the powers of the derivatives. First, we check if the equation is a polynomial in its derivatives. The terms in the equation are \(\frac{{{d}^{3}}y}{d{{x}^{3}}}\), \({\left( \frac{dy}{dx} \right)}^{2}\), and \(-{{x}^{2}}\left( \frac{{{d}^{4}}y}{d{{x}^{4}}} \right)\). All these terms involve derivatives raised to integer powers (1 for \(\frac{{{d}^{3}}y}{d{{x}^{3}}}\) and \(\frac{{{d}^{4}}y}{d{{x}^{4}}}\), and 2 for \(\frac{dy}{dx}\)). There are no terms involving transcendental functions of derivatives (like \(\sin(\frac{dy}{dx})\) or \(e^{\frac{d^2y}{dx^2}})\) or derivatives under radicals or fractions. Thus, the differential equation is a polynomial in its derivatives.
The degree is the power of the highest order derivative, which is \(\frac{{{d}^{4}}y}{d{{x}^{4}}}\).
Let's rewrite the equation to clearly see the power of the highest order derivative:
\(-{{x}^{2}}\left( \frac{{{d}^{4}}y}{d{{x}^{4}}} \right) + \frac{{{d}^{3}}y}{d{{x}^{3}}} + {{\left( \frac{dy}{dx} \right)}^{2}} = 0\)
The term with the highest order derivative is \(-{{x}^{2}}\left( \frac{{{d}^{4}}y}{d{{x}^{4}}} \right)\). The derivative itself is \(\frac{{{d}^{4}}y}{d{{x}^{4}}}\), and its power in this term is 1.
Since the equation is a polynomial in derivatives, and the power of the highest order derivative (\(\frac{{{d}^{4}}y}{d{{x}^{4}}}\)) is 1, the degree of the differential equation is 1.
| Highest Order Derivative | Power | Degree |
|---|---|---|
| \(\frac{{{d}^{4}}y}{d{{x}^{4}}}\) | 1 | 1 |
Conclusion on Degree
The order of the differential equation is 4, and its degree is 1. Therefore, the correct degree of the given differential equation is 1.
Revision Table: Order and Degree Concepts
| Concept | Definition | How to Find | Example |
|---|---|---|---|
| Order | The order of the highest derivative present in the differential equation. | Identify all derivatives and find the maximum order. | For \(\left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} = 0\), the highest order is 2. Order is 2. |
| Degree | The power of the highest order derivative when the equation is a polynomial in derivatives, free from radicals and fractions of derivatives. Undefined if not a polynomial in derivatives. | Find the order. If the equation is a polynomial in derivatives, find the power of the highest order derivative term. | For \(\left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} = 0\), order is 2. Power of \(\frac{d^2y}{dx^2}\) is 3. Degree is 3. |
Additional Information on Degree Calculation
It is crucial to remember the conditions under which the degree of a differential equation is defined. The equation must be expressible as a polynomial in the derivatives. If the equation contains terms like \(\sin(\frac{dy}{dx})\), \(\cos(\frac{d^2y}{dx^2})\), \(e^{\frac{dy}{dx}}\), \(\ln(\frac{d^3y}{dx^3})\), or derivatives under root signs that cannot be removed by squaring etc., then the degree is not defined.
For instance, the differential equation \(\frac{d^2y}{dx^2} + \sin(\frac{dy}{dx}) = 0\) has order 2, but its degree is undefined because of the \(\sin(\frac{dy}{dx})\) term, which makes it non-polynomial in derivatives.
In the given problem, \(\frac{{{d}^{3}}y}{d{{x}^{3}}}+{{\left( \frac{dy}{dx} \right)}^{2}}-{{x}^{2}}\left( \frac{{{d}^{4}}y}{d{{x}^{4}}} \right)=0\), all terms involve derivatives raised to integer powers. The coefficients (\(1\), \(1\), \(-x^2\)) do not affect whether it's a polynomial in the derivatives themselves. Since it is a polynomial in derivatives, the degree is well-defined as the power of the highest order derivative, \(\frac{{{d}^{4}}y}{d{{x}^{4}}}\), which is 1.
Similar Questions
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Important Questions from Order and Degree of a Differential Equation
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