Question

Degree of differential equation \(\sqrt[3]{\left(\frac{dy}{dx}\right)^{2}+5}=\frac{1+\left(\frac{dx}{dy}\right)^{3}}{\sqrt{4+\left(\frac{dy}{dx}\right)^{5}}}\) is

• A. \(37\)
• B. \(19\)
• C. \(10\)
• D. \(5\)

Hint:

Clear all radicals and fractions by writing p for dy/dx, then read the highest power of p.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The degree is the power of the highest order derivative after removing all radicals and fractions. Let \(p=\frac{dy}{dx}\).

Step 2: Rewrite
The equation is \(\sqrt[3]{p^{2}+5}=\frac{1+p^{-3}}{\sqrt{4+p^{5}}}\), that is \(\sqrt[3]{p^{2}+5}\,\sqrt{4+p^{5}}\,p^{3}=p^{3}+1\).

Step 3: Remove the radicals
Raise both sides to the power \(6\): \((p^{2}+5)^{2}(4+p^{5})^{3}p^{18}=(p^{3}+1)^{6}\).
The highest power of \(p\) on the left is \(4+15+18=37\), and on the right it is \(18\).

Final Answer:
The degree is \(37\). \[ \boxed{\text{(A) }37} \]