Question

The differential equation \(\left|\frac{dy}{dx}\right|+|y|+3=0\) admits

• A. infinite number of solutions
• B. no solution
• C. a unique solution
• D. many solutions

Hint:

If an expression contains \(|A|+|B|+c\) with \(c>0\), it can never become zero.

Answer 1

The Correct Option is B

Step 1: Analyze the given equation.
\[ \left|\frac{dy}{dx}\right|+|y|+3=0 \]
Step 2: Use property of modulus.
For any real number \(u\):
\[ |u|\ge 0 \]
So:
\[ \left|\frac{dy}{dx}\right|\ge 0,\quad |y|\ge 0 \]
Step 3: Minimum possible value of LHS.
\[ \left|\frac{dy}{dx}\right|+|y|+3 \ge 0+0+3=3 \]
Step 4: Can it ever be 0?
No, because LHS is always \(\ge 3\).
So equation cannot be satisfied for any \(y(x)\).
Final Answer:
\[ \boxed{\text{No solution}} \]