Q.Find the general solution of the differential equation: $x(e^{2y}-1)dy+(x^2-1)e^y\,dx=0$

Order and Degree of Differential Equations

The Intuition: What Are We Counting?

A differential equation is an equation that involves derivatives — rates of change. When you see something like

or

you are looking at relationships between a function and its derivatives. The order and degree are two numbers that classify such equations. Think of them as the "size" and "shape" of the derivative information.

Order tells you the highest number of times the function has been differentiated. If the highest derivative is $\frac{dy}{dx}$ (first derivative), the order is 1. If it's $\frac{d^2y}{dx^2}$ (second derivative), the order is 2. Simple.

Degree is trickier. It tells you the power to which the highest-order derivative is raised — after the equation has been cleared of radicals and fractions involving derivatives. If the highest derivative appears squared, the degree is 2. If it appears inside a square root, you must first remove that root before declaring the degree.

The Precise Definitions

The "provided" part is critical. You cannot read the degree directly from a messy equation — you must first rewrite it so that every derivative appears with a whole-number exponent.

Examples That Build Understanding

Example 1: $\frac{dy}{dx} + y = x$

Highest derivative is $\frac{dy}{dx}$ (first derivative). Order = 1. That derivative appears to the power 1. Degree = 1.

Example 2: $\left(\frac{d^2y}{dx^2}\right)^3 + 5\frac{dy}{dx} = \sin x$

Highest derivative is $\frac{d^2y}{dx^2}$ (second derivative). Order = 2. That derivative is raised to the power 3. Degree = 3.

Example 3: $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \frac{d^2y}{dx^2}$

Here the highest derivative is $\frac{d^2y}{dx^2}$ (order 2). But the equation is not a polynomial in derivatives — there is a square root. To find the degree, square both sides:

Now the highest derivative $\frac{d^2y}{dx^2}$ appears with power 2. Degree = 2.

Example 4: $\frac{d^2y}{dx^2} = \sin\left(\frac{dy}{dx}\right)$

Order = 2. But the equation contains $\sin$ of a derivative — it is not a polynomial in derivatives at all. Degree is not defined for such equations. Many exam questions test exactly this: if the derivative appears inside a trigonometric, logarithmic, or exponential function, the degree is simply not defined.

A Quick Reference Table

| Equation | Order | Degree | Notes |

|---|---|---|---| …