If the general solution of a differential equation is y 2+ 2cy - cx + c 2= 0, where c is an arbitrary constant, then what is the order of the differential equation?
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Understanding the Order of a Differential Equation
The order of a differential equation is the highest order of the derivative appearing in the equation. However, when given the general solution of a differential equation, there is a direct relationship between the number of arbitrary constants in the general solution and the order of the differential equation.
Specifically, the number of essential arbitrary constants in the general solution of a differential equation is equal to the order of the differential equation.
Analyzing the Given General Solution
The provided general solution is:
\(y^2 + 2cy - cx + c^2 = 0\)
In this equation, \(y\) and \(x\) are variables, and \(c\) is an arbitrary constant.
Identifying the Arbitrary Constants
The term "arbitrary constant" refers to a constant that can take any real value and is introduced during the process of integration when solving a differential equation. In the given general solution, the symbol \(c\) represents an arbitrary constant.
Let's count the number of arbitrary constants in the expression \(y^2 + 2cy - cx + c^2 = 0\). There is only one symbol, \(c\), that is explicitly mentioned as an arbitrary constant.
Determining the Order of the Differential Equation
Since the general solution contains exactly one arbitrary constant (\(c\)), the order of the corresponding differential equation must be equal to the number of these arbitrary constants.
Number of arbitrary constants = 1.
Therefore, the order of the differential equation is 1.
Conclusion
Based on the principle that the number of arbitrary constants in the general solution equals the order of the differential equation, the order of the differential equation whose general solution is \(y^2 + 2cy - cx + c^2 = 0\) is 1.
| Concept | Description |
|---|---|
| Order of DE | Highest derivative order in the equation. |
| General Solution | Solution containing arbitrary constants. |
| Arbitrary Constant | Constant introduced during integration. |
| Relationship | Number of arbitrary constants in general solution = Order of the DE. |
Revision Table: Differential Equation Concepts
- Order: Highest derivative.
- Degree: Power of the highest derivative (after rationalizing).
- General Solution: Contains arbitrary constants.
- Particular Solution: Obtained from general solution by assigning specific values to arbitrary constants (often using initial/boundary conditions).
Additional Information: Forming the Differential Equation
To find the differential equation from the general solution \(y^2 + 2cy - cx + c^2 = 0\), we would need to eliminate the arbitrary constant \(c\). This is done by differentiating the equation with respect to \(x\) and then substituting \(c\) from the original equation or the differentiated equation.
Given: \(y^2 + 2cy - cx + c^2 = 0\)
Differentiating with respect to \(x\):
\(\frac{d}{dx}(y^2) + \frac{d}{dx}(2cy) - \frac{d}{dx}(cx) + \frac{d}{dx}(c^2) = 0\)
\(2y \frac{dy}{dx} + 2c \frac{dy}{dx} - c(1) + 0 = 0\)
\((2y + 2c)\frac{dy}{dx} - c = 0\)
This equation still contains \(c\). To eliminate \(c\), we could solve for \(c\) from the original equation (which is a quadratic in \(c\)) and substitute, or solve for \(c\) from the differentiated equation: \(c = \frac{(2y + 2c)\frac{dy}{dx}}{1}\) which doesn't help directly. A better approach from the differentiated equation is to isolate \(c\):
\(c = (2y + 2c) \frac{dy}{dx}\)
This still has \(c\) on both sides. Let's rewrite the differentiated equation: \(c = (2y + 2c) y'\). This doesn't look straightforward to eliminate \(c\).
Let's go back to \( (2y + 2c)y' - c = 0 \). We can write \(c = \frac{2yy'}{1 - 2y'}\). Substituting this back into the original equation \(y^2 + 2cy - cx + c^2 = 0\) would give the differential equation. This process confirms that eliminating one arbitrary constant leads to a first-order differential equation.
Similar Questions
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