Question

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

• A. $y (1)= e ^{-\frac{1}{2}}-1$
• B. $y (1)= e ^{\frac{1}{2}}- e ^{-\frac{1}{2}}$
• C. $y (1)=1$
• D. $y (1)= e ^{\frac{1}{2}}-1$

Answer 1

Answer: (A)

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