Question

The order of the differential equation \( \frac{d}{dz}\left[\left(\frac{dy}{dz}\right)^3\right]=0 \) is

• A. not defined
• B. 1
• C. 2
• D. 3

Hint:

Even if derivatives appear in powers, the order depends only on the highest derivative after simplification, not on its power.

Answer 1

The Correct Option is C

Step 1: Recall the definition of order.
The order of a differential equation is the order of the highest derivative present in the equation.

Step 2: Write the given equation clearly.
The given equation is:
\[ \frac{d}{dz}\left[\left(\frac{dy}{dz}\right)^3\right]=0. \]

Step 3: Simplify the expression.
Let us denote \( \frac{dy}{dz} = y' \). Then the equation becomes:
\[ \frac{d}{dz}(y'^3)=0. \]

Step 4: Differentiate using chain rule.
Using the chain rule:
\[ \frac{d}{dz}(y'^3)=3(y')^2 \cdot \frac{d^2y}{dz^2}. \]
So the equation becomes:
\[ 3\left(\frac{dy}{dz}\right)^2 \cdot \frac{d^2y}{dz^2}=0. \]

Step 5: Identify the highest order derivative.
Here we observe that the highest derivative present is:
\[ \frac{d^2y}{dz^2}. \]

Step 6: Determine the order.
Since the highest derivative is of order 2, the order of the differential equation is 2.

Step 7: Final conclusion.
Thus, the order of the given differential equation is 2.
Final Answer:
\[ \boxed{2}. \]