Q.49 The order of differential equation 𝑑3𝑦/dx3+2d2y/dx2-3dy/dx+6x4y=0 is ______.
Order of Differential Equation d³y/dx³ + 2d²y/dx² − 3dy/dx + 6x⁴y = 0 Explained
The order of the differential equation d³y/dx³ + 2d²y/dx² − 3dy/dx + 6x⁴y = 0 is 3.
Concept of Order of a Differential Equation
The order of a differential equation is the order of the highest derivative of the dependent variable present in the equation.
For example, if the highest derivative is d²y/dx², the equation is second order; if the highest derivative is d³y/dx³, it is third order.
Step-by-Step Solution for the Given Equation
Consider the equation:
d³y/dx³ + 2d²y/dx² − 3dy/dx + 6x⁴y = 0
List All Derivatives Present
- Third derivative: d³y/dx³
- Second derivative: d²y/dx²
- First derivative: dy/dx
Identify the Highest Order Derivative
Among the first, second, and third derivatives, the third derivative d³y/dx³ has the highest order, which is 3.
State the Order
Therefore, the order of the differential equation is 3, because order equals the order of this highest derivative.
Note: Since none of the derivatives are raised to a power greater than 1, the degree of this equation (if asked) would be 1, but the question here specifically asks only for the order.
Explanation of Typical Options (MCQ Style)
In exams, this question usually comes with options like:
| Option | Proposed Order | Explanation |
|---|---|---|
| (A) | 1 (First order) | Incorrect: Equation contains second and third derivatives, so order cannot be 1. |
| (B) | 2 (Second order) | Incorrect: Presence of d³y/dx³ makes the equation higher than second order. |
| (C) | 3 (Third order) | Correct: Highest derivative is d³y/dx³, so order is 3. |
| (D) | 4 (Fourth order) | Incorrect: No fourth derivative like d⁴y/dx⁴ appears in the equation. |
Thus, the correct option is 3 (third order).
