Question:
The order and degree of the following differential equation are \( m \) and \( n \), respectively.
\[ \frac{\partial^3 \varphi}{\partial x^3} + \frac{\partial^2 \varphi}{\partial y^2}\frac{\partial \varphi}{\partial x} + \left(\frac{\partial^2 \varphi}{\partial x^2}\right)^2 + \frac{\partial \varphi}{\partial y} = 0 \]
The value of \( (m-n) \) is
The order and degree of the following differential equation are \( m \) and \( n \), respectively.
\[ \frac{\partial^3 \varphi}{\partial x^3} + \frac{\partial^2 \varphi}{\partial y^2}\frac{\partial \varphi}{\partial x} + \left(\frac{\partial^2 \varphi}{\partial x^2}\right)^2 + \frac{\partial \varphi}{\partial y} = 0 \]
The value of \( (m-n) \) is
\[ \frac{\partial^3 \varphi}{\partial x^3} + \frac{\partial^2 \varphi}{\partial y^2}\frac{\partial \varphi}{\partial x} + \left(\frac{\partial^2 \varphi}{\partial x^2}\right)^2 + \frac{\partial \varphi}{\partial y} = 0 \]
The value of \( (m-n) \) is
Show Hint
Find the highest derivative order first, then check the power of only that highest order term for the degree.
Updated On: Jul 27, 2026
- \( 2 \)
- \( 3 \)
- \( 1 \)
- \( 0 \)
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The Correct Option is A
Solution and Explanation
Step 1: Find the order.
The order of a differential equation is the highest derivative that appears in it.
Here the term \( \dfrac{\partial^3 \varphi}{\partial x^3} \) is a third order derivative, and no higher order term is present.
So the order is \( m = 3 \).
Step 2: Find the degree.
The degree is the power of the highest order derivative term, once the equation is written as a polynomial in the derivatives with no fractional or negative powers on any derivative.
The highest order term, \( \dfrac{\partial^3 \varphi}{\partial x^3} \), shows up with power 1 only; the squared term \( \left(\dfrac{\partial^2 \varphi}{\partial x^2}\right)^2 \) uses a lower order (second order) derivative, so it does not affect the degree count.
So the degree is \( n = 1 \).
Step 3: Compute \( m - n \).
\[ m - n = 3 - 1 = 2 \]
Final Answer:
The order is 3 and the degree is 1, giving a difference of 2. \[ \boxed{m - n = 2} \]
The order of a differential equation is the highest derivative that appears in it.
Here the term \( \dfrac{\partial^3 \varphi}{\partial x^3} \) is a third order derivative, and no higher order term is present.
So the order is \( m = 3 \).
Step 2: Find the degree.
The degree is the power of the highest order derivative term, once the equation is written as a polynomial in the derivatives with no fractional or negative powers on any derivative.
The highest order term, \( \dfrac{\partial^3 \varphi}{\partial x^3} \), shows up with power 1 only; the squared term \( \left(\dfrac{\partial^2 \varphi}{\partial x^2}\right)^2 \) uses a lower order (second order) derivative, so it does not affect the degree count.
So the degree is \( n = 1 \).
Step 3: Compute \( m - n \).
\[ m - n = 3 - 1 = 2 \]
Final Answer:
The order is 3 and the degree is 1, giving a difference of 2. \[ \boxed{m - n = 2} \]
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