Question

\(p\) and \(q\) are positive integers and \(n\lt r\lt m\). If the order and degree of the differential equation \[ \left( \frac{d^my}{dx^m}+\frac{d^ny}{dx^n} \right)^{p/q} = 5\frac{d^ry}{dx^r} \] are respectively \(4\) and \(3\), then

• A. \(n=4,\ q=3\)
• B. \(m=4,\ q=3\)
• C. \(r=4,\ q=3\)
• D. \(m=4,\ p=3\)

Hint:

Order is the order of the highest derivative present, while degree is the power of the highest order derivative after removing radicals and fractional powers.

Answer 1

The Correct Option is D

Step 1: Recall definitions.
Order = order of the highest derivative present. Degree = exponent of the highest derivative after clearing all fractional powers in the derivative terms.
Step 2: Identify the order.
The ODE $ \left(\frac{d^m y}{dx^m}+\frac{d^n y}{dx^n}\right)^{p/q}=5\frac{d^r y}{dx^r} $ has $ n<r<m $. The highest derivative is the $ m $-th. So Order $ =m $. Given order is 4, so $ m=4 $.
Step 3: Verify the ordering constraint.
With $ m=4 $, we need $ n<r<4 $, e.g., $ n=1,r=2 $. The values of $ n,r $ do not affect order or degree.
Step 4: Clear the fractional exponent to find degree.
Raise both sides to power $ q $: \[ \left(\frac{d^m y}{dx^m}+\frac{d^n y}{dx^n}\right)^p=5^q\left(\frac{d^r y}{dx^r}\right)^q \] Now the highest derivative appears with power $ p $.
Step 5: Read the degree.
Degree $ =p $. Given degree is 3, so $ p=3 $.
Step 6: State the values.
\[ m=4,\quad p=3 \] \[ \boxed{m=4,\ p=3} \]