Consider the following in respect of the differential equation: \(\frac{{{d^2}y}}{{d{x^2}}} + 2{\left( {\frac{{dy}}{{dx}}} \right)^2} + 9y = x\) 1. The degree of the differential equation is 1. 2. The order of the differential equation is 2. Which of the above statements is/are correct?
Both 1 and 2
Understanding Differential Equations: Order and Degree
A differential equation is an equation that contains an unknown function and one or more of its derivatives with respect to one or more independent variables. When analyzing a differential equation, two important properties are its order and its degree.
Let's consider the given differential equation:
\[\frac{{{d^2}y}}{{d{x^2}}} + 2{\left( {\frac{{dy}}{{dx}}} \right)^2} + 9y = x\]To determine the order and degree, we need to identify the highest order derivative present in the equation and its power.
Determining the Order of the Differential Equation
The order of a differential equation is defined by the order of the highest derivative appearing in the equation. In the given equation, the derivatives present are:
\(\frac{{dy}}{{dx}}\) (first order derivative)
\(\frac{{{d^2}y}}{{d{x^2}}}\) (second order derivative)
The highest order derivative present is \(\frac{{{d^2}y}}{{d{x^2}}}\), which is a second order derivative. Therefore, the order of this differential equation is 2.
Determining the Degree of the Differential Equation
The degree of a differential equation is the power of the highest order derivative, provided that the differential equation can be written as a polynomial in the derivatives. If the equation is not a polynomial in the derivatives (e.g., involves terms like \(\sin(\frac{dy}{dx})\) or \(\ln(\frac{d^2y}{dx^2})\)), the degree is undefined.
Let's look at the given equation again:
\[\frac{{{d^2}y}}{{d{x^2}}} + 2{\left( {\frac{{dy}}{{dx}} \right)^2} + 9y = x\]This equation is a polynomial in the derivatives \(\frac{{{d^2}y}}{{d{x^2}}}\) and \(\frac{{dy}}{{dx}}\). The highest order derivative is \(\frac{{{d^2}y}}{{d{x^2}}}\). We need to find the power to which this highest order derivative is raised.
In the term \(\frac{{{d^2}y}}{{d{x^2}}}\), the power is 1. Although the term \({\left( {\frac{{dy}}{{dx}}} \right)^2}\) involves a power of 2, this is a lower order derivative, and its power does not determine the degree of the equation. The degree is determined solely by the power of the highest order derivative.
Therefore, the degree of this differential equation is 1.
Analyzing the Statements
Let's examine the given statements based on our findings:
The degree of the differential equation is 1. Our analysis shows the degree is 1. So, this statement is correct.
The order of the differential equation is 2. Our analysis shows the order is 2. So, this statement is correct.
Both statements are correct.
| Property | Value for \(\frac{{{d^2}y}}{{d{x^2}}} + 2{\left( {\frac{{dy}}{{dx}} \right)^2} + 9y = x\) |
|---|---|
| Highest Order Derivative | \(\frac{{{d^2}y}}{{d{x^2}}}\) |
| Order | 2 |
| Power of Highest Order Derivative | 1 |
| Equation is polynomial in derivatives? | Yes |
| Degree | 1 |
Conclusion
Based on the analysis, both statements regarding the order and degree of the differential equation are correct.
Revision Table: Order and Degree of Differential Equations
| Concept | Definition | Example |
|---|---|---|
| Order | The order of the highest derivative present in the equation. | For \(y'' + (y')^3 = x\), the highest derivative is \(y''\) (second order), so the order is 2. |
| Degree | The power of the highest order derivative, provided the equation is a polynomial in derivatives. Undefined otherwise. | For \(y'' + (y')^3 = x\), the highest order derivative \(y''\) has a power of 1. The equation is a polynomial in derivatives. The degree is 1. |
| Degree (Example 2) | For \((y'')^3 + (y')^2 + y = 0\), the highest order derivative is \(y''\). Its power is 3. The equation is a polynomial in derivatives. The degree is 3. | |
| Degree (Example 3) | For \(\sin(y') + y'' = 0\), the highest order derivative is \(y''\). The term \(\sin(y')\) is not a polynomial in derivatives. The degree is undefined. |
Additional Information: Importance of Order and Degree
The order and degree of a differential equation provide crucial information about its nature and potential solution methods.
Order: The order tells us how many initial or boundary conditions are typically required to find a unique solution to the equation. A first-order equation usually needs one condition, a second-order equation needs two, and so on.
Degree: The degree can sometimes indicate the complexity or the number of possible solutions (although finding general solutions is more complex than just looking at the degree). Linear differential equations are always of degree one. Non-linear equations have a degree greater than one or an undefined degree.
Classification: Differential equations are often classified by their order (e.g., first-order ODE, second-order PDE) and linearity. Linearity depends on whether the dependent variable and its derivatives appear only in the first power and are not multiplied together. The given equation \(\frac{{{d^2}y}}{{d{x^2}}} + 2{\left( {\frac{{dy}}{{dx}}} \right)^2} + 9y = x\) is second order but non-linear because of the \({\left( {\frac{{dy}}{{dx}}} \right)^2}\) term.
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