The degree of the differential equation \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {\rm{x}} = {\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^{ - 4}}\) is
5
Understanding the Degree of a Differential Equation
To find the degree of a differential equation, we first need to understand what order and degree mean in this context. The order of a differential equation is the order of the highest derivative appearing in the equation. The degree of a differential equation is the power of the highest order derivative, provided the equation can be written as a polynomial in the derivatives. If the equation involves radicals or negative powers of derivatives, we must first clear them to express the equation as a polynomial in its derivatives.
Analyzing the Given Differential Equation
The given differential equation is:
\(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {\rm{x}} = {\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^{ - 4}}\)
We can see that the highest order derivative present is \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\), which is a first-order derivative. Thus, the order of this differential equation is 1.
Simplifying to Find the Degree
The equation involves a negative power of a term containing the derivative. To find the degree, we must eliminate this negative power and express the equation as a polynomial in the derivatives. The term \({\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^{ - 4}}\) can be written as \(\frac{1}{{\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4}}\).
So the equation becomes:
\(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {\rm{x}} = \frac{1}{{\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4}}\)
To clear the fraction involving the derivative term, we multiply both sides of the equation by \({\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4}\):
\(\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {\rm{x}}} \right){\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4} = 1\)
Determining the Highest Power of the Highest Order Derivative
The equation is now in a form where we can identify the degree. The highest order derivative is \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\). We need to find the highest power of \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) in the expanded form of this equation.
Let's look at the term \({\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4}\). When expanded using the binomial theorem, the terms will involve powers of \({\rm{y}}\) and \(-{\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\). The term with the highest power of \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) in this expansion comes from \(\left( {-{\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4\), which is \((-{\rm{x}})^4 \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^4 = {\rm{x}}^4 \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^4\).
Now consider the entire left side of the equation: \(\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {\rm{x}}} \right){\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4}\).
When we multiply out this expression, the term with the highest power of \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) will be obtained by multiplying the term with the highest power of \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) from each factor.
- From the first factor \(\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {\rm{x}}} \right)\), the term with the highest power of \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) is \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) (power 1).
- From the second factor \({\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4}\), the term with the highest power of \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) is \({\rm{x}}^4 \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^4\) (power 4).
Multiplying these terms gives: \(\left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^1 \times {\rm{x}}^4 \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^4 = {\rm{x}}^4 \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^{1+4} = {\rm{x}}^4 \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^5\).
The highest power of the highest order derivative (\(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\)) in the polynomial form of the equation is 5.
Conclusion on Degree
Therefore, the degree of the given differential equation is 5.
| Property | Value |
|---|---|
| Given Equation | \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {\rm{x}} = {\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^{ - 4}}\) |
| Simplified Form (Polynomial in derivatives) | \(\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {\rm{x}}} \right){\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4} = 1\) |
| Highest Order Derivative | \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) |
| Order | 1 |
| Highest Power of Highest Order Derivative | 5 |
| Degree | 5 |
Revision Table: Order and Degree of Differential Equations
| Concept | Definition | Example |
|---|---|---|
| Order | The order of the highest derivative in the equation. | \(\frac{{{\rm{d}}^2{\rm{y}}}}{{{\rm{dx}}^2}} + \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^3 + {\rm{y}} = 0\). Order is 2. |
| Degree | The power of the highest order derivative after the equation is made free from radicals and fractions involving derivatives, and is written as a polynomial in derivatives. | For \(\frac{{{\rm{d}}^2{\rm{y}}}}{{{\rm{dx}}^2}} + \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^3 + {\rm{y}} = 0\), the degree is 1 (power of \(\frac{{{\rm{d}}^2{\rm{y}}}}{{{\rm{dx}}^2}}\)). For \(\left(1 + \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^2\right)^{3/2} = \frac{{{\rm{d}}^2{\rm{y}}}}{{{\rm{dx}}^2}}\), squaring both sides gives \(\left(1 + \left(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\right)^2\right)^{3} = \left(\frac{{{\rm{d}}^2{\rm{y}}}}{{{\rm{dx}}^2}}\right)^2\). The highest order is 2 (\(\frac{{{\rm{d}}^2{\rm{y}}}}{{{\rm{dx}}^2}}\)) and its power is 2. Degree is 2. |
Additional Information on Differential Equations
Differential equations are mathematical equations that relate a function with its derivatives. They are fundamental in physics, engineering, biology, economics, and many other fields because they describe how quantities change over time or space.
- Ordinary Differential Equations (ODEs): These involve functions of a single independent variable and their derivatives. The given problem involves an ODE because \({\rm{y}}\) is a function of a single variable \({\rm{x}}\), and the equation involves only ordinary derivatives (\(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\)).
- Partial Differential Equations (PDEs): These involve functions of multiple independent variables and their partial derivatives.
- Linear vs. Non-linear: A differential equation is linear if the dependent variable and its derivatives appear only in the first power and are not multiplied together. Otherwise, it is non-linear. The given equation is non-linear due to the term \({\left( {{\rm{y}} - {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^4}\) involving powers and products of \({\rm{y}}\) and \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\).
Understanding the order and degree is often the first step in classifying a differential equation, which helps determine the methods suitable for solving it.
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