Could someone solve these:
xyy'+y^2=sinx by letting y^2=v
y'=2/(x+2y-3) by letting x+2y-3=v
2y'y''=1+(y')^2 by letting y'=v
or explain one and if they are similar then i'd be fine?
Could someone solve these:
xyy'+y^2=sinx by letting y^2=v
y'=2/(x+2y-3) by letting x+2y-3=v
2y'y''=1+(y')^2 by letting y'=v
or explain one and if they are similar then i'd be fine?
Hint:
$\displaystyle 2y'y''=1+(y')^2$
$\displaystyle \dfrac{2y'y''}{1+(y')^2}=1$
$\displaystyle \Bigl(\ln(1+(y')^2)\Bigl)'=1$
$\displaystyle \ln(1+(y')^2)=x+C$
$\displaystyle 1+(y')^2=e^{x+C}=C_1e^x$
$\displaystyle (y')^2=C_1e^x-1$
$\displaystyle y'=\pm\sqrt{C_1e^x-1}$
$\displaystyle y=\pm\int\!\sqrt{C_1e^x-1}\,dx$