Order And Degree Of Differential Equation

Concept: Calculus - Formation of Differential Equations (Eliminating Arbitrary Constants).

Step 1: Differentiate the given equation once. The given equation is . Differentiate with respect to using the product rule ( ): .

Step 2: Simplify by substituting the original function. Notice that the first term in our derivative, , is exactly equal to our original function . Substitute back into the equation: . Let's call this Equation (i).

Step 3: Differentiate a second time. Differentiate Equation (i) with respect to to get the second derivative: .

Step 4: Eliminate constants using previous equations. Rearrange Equation (i) to isolate the exponential term: . Also, notice that the last term, , is exactly the negative of our original function , so it equals . Substitute both of these findings into our second derivative equation:
.

Step 5: Rearrange into the final standard form. Combine the like terms on the right side of the equation: . Move all terms to the left side to match the standard format of a differential equation: . $ $

Concept: The order of a differential equation is defined as the highest order derivative present in the equation. In general:
• First derivative → first order
• Second derivative → second order
• Third derivative → third order The order depends only on the highest derivative and not on its power.

Step 1: Identify all derivatives present in the equation. Given differential equation: Here we observe two derivatives:
• First derivative:
• Second derivative:

Step 2: Determine the highest order derivative. The highest order derivative appearing in the equation is: This is a second order derivative.

Step 3: Note about powers of derivatives. Even though contains a power of 3, the order is determined only by the derivative itself, not by its exponent. Therefore, the order of the differential equation is