What are the order and degree respectively of the differential equation whose solution is y = cx + c 2– 3c 3/2 + 2, where c is a parameter?
1, 4
Finding the Order and Degree of a Differential Equation
The question asks for the order and degree of the differential equation whose general solution is given. The general solution is provided as \(y = cx + c^2 - 3c^{3/2} + 2\), where \(c\) is a parameter.
Eliminating the Parameter to Find the Differential Equation
To obtain the differential equation from a general solution involving a parameter, we need to eliminate the parameter by differentiation. Here, the parameter is \(c\). We differentiate the given solution with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}(cx + c^2 - 3c^{3/2} + 2) \]Since \(c\) is a parameter, it is treated as a constant with respect to \(x\). Differentiating each term:
\[ \frac{dy}{dx} = c \frac{d}{dx}(x) + \frac{d}{dx}(c^2) - 3 \frac{d}{dx}(c^{3/2}) + \frac{d}{dx}(2) \] \[ \frac{dy}{dx} = c(1) + 0 - 3(0) + 0 \] \[ \frac{dy}{dx} = c \]So, we have found that \(c = \frac{dy}{dx}\). Let's denote \(\frac{dy}{dx}\) as \(y'\) for simplicity. Thus, \(c = y'\).
Now, substitute this expression for \(c\) back into the original general solution equation:
\[ y = (y')x + (y')^2 - 3(y')^{3/2} + 2 \]This equation relates \(y\), \(x\), and \(y'\). This is the differential equation.
Determining the Order of the Differential Equation
The order of a differential equation is the order of the highest derivative present in the equation. In the equation \(y = y'x + (y')^2 - 3(y')^{3/2} + 2\), the highest derivative is \(y' = \frac{dy}{dx}\), which is the first derivative.
Therefore, the order of this differential equation is 1.
Determining the Degree of the Differential Equation
The degree of a differential equation is the power of the highest order derivative when the differential equation is expressed as a polynomial in its derivatives. The equation we obtained is \(y = y'x + (y')^2 - 3(y')^{3/2} + 2\).
This equation contains a term with a fractional power of \(y'\), which is \(3(y')^{3/2}\). To express the equation as a polynomial in derivatives, we need to eliminate this fractional power. We can do this by isolating the term and squaring both sides.
Rearrange the equation to isolate the term with the fractional power:
\[ 3(y')^{3/2} = y'x + (y')^2 - y + 2 \]Now, square both sides of the equation:
\[ (3(y')^{3/2})^2 = (y'x + (y')^2 - y + 2)^2 \] \[ 3^2 \cdot (y')^{(3/2) \cdot 2} = (y'x + (y')^2 - y + 2)^2 \] \[ 9 (y')^3 = (y'x + (y')^2 - y + 2)^2 \]This differential equation, \(9 (y')^3 = (y'x + (y')^2 - y + 2)^2\), is now a polynomial in the derivative \(y'\). The highest order derivative is \(y'\) (first order).
To find the degree, we look at the highest power of the highest order derivative (\(y'\)) in this polynomial equation.
- The left side is \(9(y')^3\). The power of \(y'\) is 3.
- The right side is \((y'x + (y')^2 - y + 2)^2\). When this expression is expanded, the term with the highest power of \(y'\) will come from squaring the term with the highest power of \(y'\) inside the parenthesis. The highest power of \(y'\) inside the parenthesis is 2 (from the term \((y')^2\)). Squaring this term gives \(((y')^2)^2 = (y')^4\).
Thus, when the entire equation is expanded and written as a polynomial in \(y'\), the highest power of \(y'\) will be 4.
Therefore, the degree of the differential equation is 4.
Conclusion
The differential equation derived from the given solution is \(y = y'x + (y')^2 - 3(y')^{3/2} + 2\), which can be written as the polynomial equation \(9(y')^3 = (y'x + (y')^2 - y + 2)^2\).
- The highest order derivative is \(y' = \frac{dy}{dx}\), so the order is 1.
- The highest power of the highest order derivative (\(y'\)) in the polynomial form is 4, so the degree is 4.
The order and degree are 1 and 4, respectively.
Revision Table: Order and Degree Concepts
| Concept | Definition | How to Find |
| Order of a DE | The order of the highest derivative appearing in the equation. | Identify all derivatives in the equation. The highest order among them is the order of the DE. |
| Degree of a DE | The power of the highest order derivative, after the equation has been made free from radicals and fractions as far as derivatives are concerned. | Ensure the DE is a polynomial in derivatives. Identify the highest order derivative and find its highest power. |
Additional Information on Differential Equations
A differential equation is an equation that relates one or more functions and their derivatives. They are used to model various processes in physics, engineering, biology, economics, and many other fields.
- General Solution: A general solution of a differential equation is a family of functions that satisfy the equation. It typically contains arbitrary constants (parameters).
- Particular Solution: A particular solution is obtained from the general solution by assigning specific values to the arbitrary constants, usually based on initial or boundary conditions.
- Formation of DE: A differential equation can be formed from a given relation (general solution) by eliminating the arbitrary constants through successive differentiation. The order of the resulting differential equation is usually equal to the number of essential arbitrary constants in the general solution. In this problem, there was one parameter \(c\), and the resulting differential equation has order 1.
The concepts of order and degree are fundamental in classifying differential equations, which helps in determining the appropriate methods for solving them.
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