Differential Equation: Which of the following is true for y(x) that satisfies the… | JEE Main 2021

JEE Main 2021 (Online) 17th March Morning ShiftQuestion 1

Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y$; $y(0)=0$ :

Differential Equation

$y(1)=e^{-\frac{1}{2}}-1$✓ Correct answer$y(1)=e^{\frac{1}{2}}-1$$y(1)=e^{\frac{1}{2}}-e^{-\frac{1}{2}}$$y(1)=1$

Solution

Concept. $xy-1+x-y=(x-1)(y+1)$; then separate. Formulas used. (1) $xy+x-y-1=(x-1)(y+1)$Step 1 - Separate. By (1), $\dfrac{dy}{y+1}=(x-1)\,dx$. Step 2 - Integrate. $\ln(y+1)=\dfrac{x^2}{2}-x+c$; $y(0)=0$ gives $c=0$. Step 3 - At $x=1$. $\ln(y+1)=-\frac12$, so $y(1)=e^{-1/2}-1$. Answer: (A)

Differential Equation - JEE Mathematics question (JEE Main 2021 (Online) 17th March Morning Shift)

More from Differential Equation