Which of the following is true for y ( x ) that satisfies the differential equation ( dy / dx )= xy -1+ x - y ; y (0)=0

Question

Which of the following is true for y ( x ) that satisfies the differential equation ( dy / dx )= xy -1+ x - y ; y (0)=0 (A) y (1)= e -(1/2)-1 (B) y (1)= e (1/2)- e -(1/2) (C) y (1)=1 (D) y (1)= e (1/2)-1

Tardigrade Admin · Accepted Answer

( dy / dx )=(1+ y )( x -1) ( dy /( y +1))=( x -1) d x Integrate ln ( y +1)=( x 2/2)- x + c (0,0) ⇒ c =0 ⇒ y = e ((x2/2)- x )-1

Q. Which of the following is true for $y (x)$ that satisfies the differential equation $\frac{d y}{d x} = x y - 1 + x - y; y (0) = 0$

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

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

$y (1) = 1$

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

$\frac{d y}{d x} = (1 + y) (x - 1)$
$\frac{d y}{( y + 1 )} = (x - 1) d x$
Integrate $ln (y + 1) = \frac{x ^{2}}{2} - x + c$
$(0, 0) \Rightarrow c = 0 \Rightarrow y = e^{(\frac{x ^{2}}{2} - x)} - 1$

Q. Which of the following is true for y(x) that satisfies the differential equation dxdy​=xy−1+x−y;y(0)=0

y(1)=e−21​−1

y(1)=e21​−e−21​

y(1)=1

y(1)=e21​−1