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
\(p\) 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
Show 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.
Updated On: Jun 25, 2026
- \(n=4,\ q=3\)
- \(m=4,\ q=3\)
- \(r=4,\ q=3\)
- \(m=4,\ p=3\)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Determine the order of the differential equation.
Given: \[ \left( \frac{d^my}{dx^m}+\frac{d^ny}{dx^n} \right)^{p/q} = 5\frac{d^ry}{dx^r} \] We are given \[ n\lt r\lt m \] The highest order derivative present is \[ \frac{d^my}{dx^m} \] Hence, order of the differential equation is \[ m \] Given order is \(4\), therefore \[ m=4 \]
Step 2: Find the degree of the differential equation.
To define degree, the equation must be free from fractional powers.
Raise both sides to the power \(q\): \[ \left( \frac{d^my}{dx^m}+\frac{d^ny}{dx^n} \right)^p = 5^q \left( \frac{d^ry}{dx^r} \right)^q \] Now the highest order derivative is \[ \frac{d^my}{dx^m} \] and its highest power is \[ p \] Thus, degree of the differential equation is \[ p \] Given degree is \(3\), hence \[ p=3 \]
Step 3: Final conclusion.
Therefore, \[ \boxed{m=4,\ p=3} \]
Given: \[ \left( \frac{d^my}{dx^m}+\frac{d^ny}{dx^n} \right)^{p/q} = 5\frac{d^ry}{dx^r} \] We are given \[ n\lt r\lt m \] The highest order derivative present is \[ \frac{d^my}{dx^m} \] Hence, order of the differential equation is \[ m \] Given order is \(4\), therefore \[ m=4 \]
Step 2: Find the degree of the differential equation.
To define degree, the equation must be free from fractional powers.
Raise both sides to the power \(q\): \[ \left( \frac{d^my}{dx^m}+\frac{d^ny}{dx^n} \right)^p = 5^q \left( \frac{d^ry}{dx^r} \right)^q \] Now the highest order derivative is \[ \frac{d^my}{dx^m} \] and its highest power is \[ p \] Thus, degree of the differential equation is \[ p \] Given degree is \(3\), hence \[ p=3 \]
Step 3: Final conclusion.
Therefore, \[ \boxed{m=4,\ p=3} \]
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