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Question #40949

Difficulty: 8/10 | 🟢 Free

The Question

Let $O_1$ and $D_1$ be the order and degree of the differential equation representing the family of curves $y = c e^x + c^3$ , where $c$ is an arbitrary real constant. Let $O_2$ and $D_2$ be the order and degree of the differential equation of the orthogonal trajectories of this family. Find the value of $O_1 + D_1 + O_2 + D_2$ .

Options

A.
6
B.
8
C.
4
D.
10

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Algebra of Derivatives|Orthogonal Trajectories|Order and Degree Definition

Why This Question Is Tricky

Common Mistakes

The question tests the fundamental definition of the order of a differential equation and its interaction with orthogonal trajectories.

A common trap is differentiating the equation twice (e.g., $y' = c e^x \implies y'' = c e^x \implies y'' = y'$ ) to eliminate the constant, incorrectly yielding an order 2, degree 1 equation ( $O_1=2, D_1=1$ ).

This happens because students forget that the order must exactly match the number of independent arbitrary constants (here, one).

If they fall for this, they will also find an incorrect orthogonal trajectory DE (which would be $y'' = -y'$ ), leading to a sum of 6 instead of the correct 8.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Algebra of Derivatives
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free