Differential Equations
Order and Degree
Grade 12
Question:
<p>If \(m\) and \(n\) are order and degree of the equation \(2\left(\frac{d^2y}{dx^2}\right)^4 + 3\frac{d^2y}{dx^2}\left(\frac{d^3y}{dx^3}\right)^5 = x^2 - 1\), then</p>
<p>(A) \(m = 3, n = 5\)</p>
<p>(B) \(m = 2, n = 4\)</p>
<p>(C) \(m = 3, n = 4\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Order is the highest derivative present; degree is the highest power of the highest order derivative.
<p>Given: $2\left(\frac{d^2y}{dx^2}\right)^4 + 3\frac{d^2y}{dx^2}\left(\frac{d^3y}{dx^3}\right)^5 = x^2 - 1$</p><p><strong>Order:</strong> The order is the highest derivative present. Here we have $\frac{d^3y}{dx^3}$ (third derivative), so order $m = 3$</p><p><strong>Degree:</strong> The degree is the highest power of the highest order derivative when the equation is polynomial. The highest order derivative is $\frac{d^3y}{dx^3}$, and its highest power is 5 (from the term $\left(\frac{d^3y}{dx^3}\right)^5$), so degree $n = 5$</p><p>Therefore, $m = 3, n = 5$</p>
Correct Answer: A