How do I solve this?Solve y′=(y−5)(y+5) if y(2)=0?I used methods of separation and did a partial fraction integral and ended up with(ln(y+5)-ln(y-5))/10 = xI'm not sure if I did this correctly nor do I know how to solve for y at this point. Please send help! Thank you.

Accepted Solution

Answer with Step-by-step explanation:The given differential euation is[tex]\frac{dy}{dx}=(y-5)(y+5)\\\\\frac{dy}{(y-5)(y+5)}=dx\\\\(\frac{A}{y-5}+\frac{B}{y+5})dy=dx\\\\\frac{1}{100}\cdot (\frac{10}{y-5}-\frac{10}{y+5})dy=dx\\\\\frac{1}{100}\cdot \int (\frac{10}{y-5}-\frac{10}{y+5})dy=\int dx\\\\10[ln(y-5)-ln(y+5)]=100x+10c\\\\ln(\frac{y-5}{y+5})=10x+c\\\\\frac{y-5}{y+5}=ke^{10x}[/tex]where 'k' is constant of integration whose value is obtained by the given condition that y(2)=0\\[tex]\frac{0-5}{0+5}=ke^{20}\\\\k=\frac{-1}{e^{20}}\\\\\therefore k=-e^{-20}[/tex]Thus the solution of the differential becomes  [tex]\frac{y-5}{y+5}=e^{10x-20}[/tex]