If \(y = y(x)\) is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = \frac{y + 2}{x + 1}\), \(x \gt -1\), satisfying \(y(0) = 1\), then \(y(2)\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 7
The equation is variables separable:
\( \displaystyle \frac{dy}{y + 2} = \frac{dx}{x + 1} \)
Integrating both sides:
\(\log|y + 2| = \log|x + 1| + \log c\)
\(y + 2 = c(x + 1)\)
Putting x = 0, y = 1:
\(3 = c \times 1,\quad c = 3\)
\(y + 2 = 3(x + 1)\)
\(y = 3x + 1\)
At x = 2:
\(y(2) = 3 \times 2 + 1 = 7\)