d3y/dx3 + 2d2y/dx2 − 3dy/dx + 6x4y = 0
is ______.
Find the Order of the Differential Equation d³y/dx³ + 2d²y/dx² − 3dy/dx + 6x⁴y = 0
Understanding the Given Differential Equation
This question asks us to determine the order of a differential equation. Finding the order is one of the most fundamental tasks in differential equations because the order tells us the highest level of differentiation of the dependent variable present in the equation.
The given differential equation is:
d3y/dx3 + 2d2y/dx2 − 3dy/dx + 6x4y = 0
At first glance, the equation contains several terms involving different derivatives of y. To determine the order, we do not need to solve the differential equation. We only need to identify all the derivatives present and find the derivative with the highest order.
What Is the Order of a Differential Equation?
The order of a differential equation is defined as the order of the highest derivative of the dependent variable appearing in that equation.
For example, if the highest derivative present in an equation is dy/dx, the differential equation is of first order. If the highest derivative is d2y/dx2, it is a second-order differential equation. Similarly, if the highest derivative is d3y/dx3, the equation is classified as a third-order differential equation.
Therefore, the basic rule is:
Order of a differential equation = Order of the highest derivative present in the equation
Step-by-Step Solution
Step 1: Write the Given Differential Equation Clearly
The given equation is:
d3y/dx3 + 2d2y/dx2 − 3dy/dx + 6x4y = 0
The equation contains the dependent variable y and several derivatives of y with respect to the independent variable x. We now examine each term separately to identify the derivative order associated with it.
Step 2: Identify the Derivatives Present in the Equation
The first derivative term is:
d3y/dx3
This represents the third derivative of y with respect to x. Therefore, its order is 3.
The next derivative term is:
2d2y/dx2
This contains the second derivative of y with respect to x. The numerical coefficient 2 does not affect the order of the derivative. Therefore, this term contains a derivative of order 2.
The third derivative-containing term is:
−3dy/dx
This contains the first derivative of y with respect to x. Again, the coefficient −3 has no effect on the order. Therefore, this term contains a derivative of order 1.
The final term is:
6x4y
This term contains y itself but does not contain any derivative of y. Therefore, it does not determine the order of the differential equation.
Step 3: Compare the Orders of All Derivatives
The derivative orders present in the equation are:
Third derivative: d3y/dx3 → Order 3
Second derivative: d2y/dx2 → Order 2
First derivative: dy/dx → Order 1
Among these derivatives, the highest-order derivative is:
d3y/dx3
Since this is the third derivative of y, the order of the complete differential equation is 3.
Step 4: State the Order of the Differential Equation
Using the definition of the order of a differential equation:
Order = Order of the highest derivative
The highest derivative in the given equation is:
d3y/dx3
Therefore:
Order = 3
Why the Term d³y/dx³ Determines the Order
The order of a differential equation depends only on the highest derivative present. It does not depend on the number of derivative terms, their coefficients, or the powers of the independent variable x appearing elsewhere in the equation.
In the given expression, d3y/dx3 is the highest derivative. Although the equation also contains the second derivative and the first derivative, these lower-order derivatives do not affect the classification of the equation.
Therefore, as soon as we identify d3y/dx3 as the highest derivative, we can conclude that the equation is a third-order differential equation.
Why the Power x⁴ Does Not Make the Order Equal to 4
The term 6x4y may sometimes cause confusion because x is raised to the fourth power. However, the order of a differential equation is never determined by the highest power of x or y.
The exponent 4 in x4 is simply an algebraic power of the independent variable x. It does not represent differentiation. Therefore, x4 has no role in determining the order of the differential equation.
The only relevant quantities are the derivatives of y. Since the highest derivative is the third derivative, the order remains 3.
Difference Between Order and Degree of a Differential Equation
Order and degree are two different properties of a differential equation. The order is determined by the highest derivative present, whereas the degree is the power of the highest-order derivative after the differential equation has been expressed as a polynomial in derivatives.
For the given equation:
d3y/dx3 + 2d2y/dx2 − 3dy/dx + 6x4y = 0
the highest-order derivative is d3y/dx3, so the order is 3. This highest-order derivative appears to the first power, so the degree of the equation is 1.
Therefore:
Order = 3
and
Degree = 1
However, since the question asks only for the order, the required answer is 3.
Classification of the Given Differential Equation
The given equation can also be classified more completely. It is an ordinary differential equation because all derivatives are taken with respect to a single independent variable x.
It is a third-order differential equation because the highest derivative is d3y/dx3. It is also linear because y and all its derivatives appear only to the first power and are not multiplied together.
Thus, the equation is a third-order linear ordinary differential equation.
Final Answer
The highest derivative present is d3y/dx3. Therefore, the order of the given differential equation is 3.
