Question

The order of the differential equation obtained by eliminating arbitrary constants in the family of curves c₁y = (c₂ + c₃)e^x+c₄ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is A

Step 1: Express the equation in a more manageable form.

The equation can be rewritten as:

$$ y = \frac{(c_2 + c_3) e^x + c_4}{c_1}. $$

This is a family of curves that depends on the constants $ c_1 $, $ c_2 $, $ c_3 $, and $ c_4 $.

Step 2: Differentiate with respect to $ x $ to eliminate arbitrary constants.

To eliminate the constants, we differentiate the equation with respect to $ x $. Each differentiation reduces the number of arbitrary constants:

• First differentiation eliminates one constant.
• Second differentiation eliminates another constant.

After two differentiations, we obtain a differential equation involving only $ y $ and its derivatives, with all the constants eliminated.

The order of the differential equation after eliminating all four constants is $ \boxed{2} $.

Answer 2

Rewrite the equation as:

$$ c_1 y = (c_2+c_3)e^x + c_4 = c_5 e^x + c_4, \quad \text{where } c_5 = c_2 + c_3. $$

Divide by $ c_1 $:

$$ y = \frac{c_5}{c_1} e^x + \frac{c_4}{c_1} = c_6 e^x + c_7, \quad \text{where } c_6 = \frac{c_5}{c_1}, \, c_7 = \frac{c_4}{c_1}. $$

This equation has two arbitrary constants ($ c_6 $ and $ c_7 $). Differentiating twice:

$$ y' = c_6 e^x, \quad y'' = c_6 e^x. $$

Thus, $ y' = y'' $, and the order of the differential equation is $ \boxed{2} $.