Question

The order and degree of the differential equation \(e^{\frac{dy}{dx}}+\frac{dy}{dx}=x\) is

• A. order = 1 and degree = 1
• B. order = 1 and degree = 2
• C. order = 1 and degree = not defined
• D. order = not defined and degree = 1

Hint:

Order is the highest derivative present; degree exists only if the equation is a polynomial in the derivatives.

Answer 1

The Correct Option is C

Step 1: Expand the exponential
$e^{dy/dx}=1+\frac{dy}{dx}+\frac{1}{2!}\left(\frac{dy}{dx}\right)^{2}+\cdots$ which has infinitely many powers.

Step 2: Polynomial test
An infinite series is not a polynomial in $\frac{dy}{dx}$, so no highest power exists.

Step 3: Result
Order is 1 and degree is not defined.

Final Answer:
The equation is not polynomial in $\frac{dy}{dx}$, so the degree is undefined. \[ \boxed{\text{(C) }\text{order = 1 and degree = not defined}} \]