4.16 Problems 751 to 800

   4.16.1 \((\sinh (x)+1) y'(x) \sinh (y(x))+\cosh (x) (\cosh (y(x))-1)=0\)
   4.16.2 \(y'(x)^2=a x^n\)
   4.16.3 \(y'(x)^2=y(x)\)
   4.16.4 \(y'(x)^2=x-y(x)\)
   4.16.5 \(y'(x)^2=x^2+y(x)\)
   4.16.6 \(x^2+y'(x)^2=4 y(x)\)
   4.16.7 \(3 x^2+y'(x)^2=8 y(x)\)
   4.16.8 \(a x^2+b y(x)+y'(x)^2=0\)
   4.16.9 \(y'(x)^2=y(x)^2+1\)
   4.16.10 \(y'(x)^2=1-y(x)^2\)
   4.16.11 \(y'(x)^2=a^2-y(x)^2\)
   4.16.12 \(y'(x)^2=a^2 y(x)^2\)
   4.16.13 \(y'(x)^2=a+b y(x)^2\)
   4.16.14 \(y'(x)^2=x^2 y(x)^2\)
   4.16.15 \(y'(x)^2=(y(x)-1) y(x)^2\)
   4.16.16 \(y'(x)^2=(y(x)-a) (y(x)-b) (y(x)-c)\)
   4.16.17 \(y'(x)^2=a^2 y(x)^n\)
   4.16.18 \(y'(x)^2=a^2 y(x)^2 \left (1-\log ^2(y(x))\right )\)
   4.16.19 \(f(x) (y(x)-a) (y(x)-b)+y'(x)^2=0\)
   4.16.20 \(f(x) (y(x)-a)^2 (y(x)-b)+y'(x)^2=0\)
   4.16.21 \(f(x) (y(x)-a) (y(x)-b) (y(x)-c)+y'(x)^2=0\)
   4.16.22 \(f(x) (y(x)-a)^2 (y(x)-b) (y(x)-c)+y'(x)^2=0\)
   4.16.23 \(f(x) (y(x)-\text {a1}) (y(x)-\text {a2}) (y(x)-\text {a3}) (y(x)-\text {a4})+y'(x)^2=0\)
   4.16.24 \(y'(x)^2=f(x)^2 (y(x)-a) (y(x)-b) (y(x)-c)^2\)
   4.16.25 \(y'(x)^2=f(x)^2 (y(x)-a) (y(x)-b) (y(x)-u(x))^2\)
   4.16.26 \(y'(x)^2+2 y'(x)+x=0\)
   4.16.27 \(a (x-y(x))+y'(x)^2-2 y'(x)=0\)
   4.16.28 \(y'(x)^2-2 y'(x)-y(x)^2=0\)
   4.16.29 \(y'(x)^2-5 y'(x)+6=0\)
   4.16.30 \(y'(x)^2-7 y'(x)+12=0\)
   4.16.31 \(a y'(x)+b+y'(x)^2=0\)
   4.16.32 \(a y'(x)+b x+y'(x)^2=0\)
   4.16.33 \(a y'(x)+b y(x)+y'(x)^2=0\)
   4.16.34 \(y'(x)^2+x y'(x)+1=0\)
   4.16.35 \(y'(x)^2+x y'(x)-y(x)=0\)
   4.16.36 \(y'(x)^2-x y'(x)+y(x)=0\)
   4.16.37 \(y'(x)^2-x y'(x)-y(x)=0\)
   4.16.38 \(y'(x)^2+x y'(x)-y(x)+x=0\)
   4.16.39 \(y'(x)^2+(1-x) y'(x)+y(x)=0\)
   4.16.40 \(y'(x)^2-(x+1) y'(x)+y(x)=0\)
   4.16.41 \(y'(x)^2-(2-x) y'(x)-y(x)+1=0\)
   4.16.42 \((a+x) y'(x)+y'(x)^2-y(x)=0\)
   4.16.43 \(y'(x)^2-2 x y'(x)+1=0\)
   4.16.44 \(-3 x^2+2 x y'(x)+y'(x)^2=0\)
   4.16.45 \(y'(x)^2+2 x y'(x)-y(x)=0\)
   4.16.46 \(y'(x)^2+2 x y'(x)-y(x)=0\)
   4.16.47 \(y'(x)^2-2 x y'(x)+2 y(x)=0\)
   4.16.48 \(y'(x)^2-(2 x+1) y'(x)-(1-x) x=0\)
   4.16.49 \(y'(x)^2+2 (1-x) y'(x)-2 (x-y(x))=0\)
   4.16.50 \(y'(x)^2+3 x y'(x)-y(x)=0\)