JEE PYQ: Differential Equation Question 3
Question 3 - 2021 (17 Mar Shift 1)
Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{dy}{dx} = xy - 1 + x - y$; $y(0) = 0$:
(1) $y(1) = e^{1/2} - 1$
(2) $y(1) = e^{1/2} - e^{-1/2}$
(3) $y(1) = 1$
(4) $y(1) = e^{1/2} - 1$
Show Answer
Answer: (1)
Solution
$\frac{dy}{dx} = (1+y)(x-1)$. Separating: $\frac{dy}{1+y} = (x-1),dx$. Integrating: $\ln(y+1) = \frac{x^2}{2} - x + c$. At $(0,0)$: $c = 0$. So $y = e^{x^2/2 - x} - 1$. $y(1) = e^{1/2-1} - 1 = e^{-1/2} - 1$. Checking options: answer is (1).
BETA
