4.15 Problems 701 to 750

   4.15.1 \(x \left (2 y(x)^3+y(x)+x\right ) y'(x)=(x-y(x)) y(x)\)
   4.15.2 \(\left (-7 x y(x)^3-y(x)+5 x\right ) y'(x)-y(x)^4+5 y(x)=0\)
   4.15.3 \(y(x) \left (1-2 x^3 y(x)\right )+x \left (1-2 x y(x)^3\right ) y'(x)=0\)
   4.15.4 \(x \left (-2 x y(x)^3-x y(x)^2+2\right ) y'(x)+2 y(x)+1=0\)
   4.15.5 \(\left (-10 x^2 y(x)^3+3 y(x)^2+2\right ) y'(x)=x \left (5 y(x)^4+1\right )\)
   4.15.6 \(x y'(x) \left (a+b x y(x)^3\right )+y(x) \left (a+c x^3 y(x)\right )=0\)
   4.15.7 \(y(x) \left (1-2 x^3 y(x)^2\right )+x \left (1-2 x^2 y(x)^3\right ) y'(x)=0\)
   4.15.8 \(x (1-x y(x)) \left (1-x^2 y(x)^2\right ) y'(x)+y(x) (x y(x)+1) \left (x^2 y(x)^2+1\right )=0\)
   4.15.9 \(\left (x^2-y(x)^4\right ) y'(x)=x y(x)\)
   4.15.10 \(\left (x^3-y(x)^4\right ) y'(x)=3 x^2 y(x)\)
   4.15.11 \(\left (a^2 x^2+\left (x^2+y(x)^2\right )^2\right ) y'(x)=a^2 x y(x)\)
   4.15.12 \(2 \left (x-y(x)^4\right ) y'(x)=y(x)\)
   4.15.13 \(\left (-2 y(x)^4-x y(x)^3+4 x\right ) y'(x)=y(x) \left (y(x)^3+2\right )\)
   4.15.14 \(y(x) y'(x) \left ((a x+b y(x))^3+a x^3\right )+x \left ((a x+b y(x))^3+b y(x)^3\right )=0\)
   4.15.15 \(\left (2 x^2 y(x)^3+x y(x)^4+2 y(x)+x\right ) y'(x)+y(x) \left (y(x)^4+1\right )=0\)
   4.15.16 \(2 x \left (x^3+y(x)^4\right ) y'(x)=y(x) \left (x^3+2 y(x)^4\right )\)
   4.15.17 \(x \left (1-x^2 y(x)^4\right ) y'(x)+y(x)=0\)
   4.15.18 \(\left (x^2-y(x)^5\right ) y'(x)=2 x y(x)\)
   4.15.19 \(x \left (x^3+y(x)^5\right ) y'(x)=y(x) \left (x^3-y(x)^5\right )\)
   4.15.20 \(y(x)^3 \left (3 x^5 y(x)^5-1\right )+x^3 \left (5 x^3 y(x)^7+1\right ) y'(x)=0\)
   4.15.21 \(y'(x) (\text {f1}(x,y(x))+x \text {f2}(x,y(x)))=y(x) \text {f2}(x,y(x))+\text {f3}(x,y(x))\)
   4.15.22 \(y'(x) (a (y(x)+x)+1)^n+a (y(x)+x)^n=0\)
   4.15.23 \(x y'(x) \left (a+x y(x)^n\right )+b y(x)=0\)
   4.15.24 \(f(x) y(x)^m y'(x)+g(x) y(x)^{m+1}+h(x) y(x)^n=0\)
   4.15.25 \(\sqrt {b^2+y(x)^2} y'(x)=\sqrt {a^2+x^2}\)
   4.15.26 \(\sqrt {b^2-y(x)^2} y'(x)=\sqrt {a^2-x^2}\)
   4.15.27 \(\sqrt {Y} y'(x)=\sqrt {X}\)
   4.15.28 \(\left (\sqrt {y(x)+x}+1\right ) y'(x)+1=0\)
   4.15.29 \(\sqrt {x y(x)} y'(x)-y(x)+x=\sqrt {x y(x)}\)
   4.15.30 \(\left (x-2 \sqrt {x y(x)}\right ) y'(x)=y(x)\)
   4.15.31 \(\left (x^2+1\right )^{3/2} \left (y(x)+\sqrt {y(x)^2+1}\right ) y'(x)=y(x)^2+1\)
   4.15.32 \(\left (x^2+1\right )^{3/2} \left (y(x)+\sqrt {y(x)^2+1}\right ) y'(x)=y(x)^2+1\)
   4.15.33 \(\left (x-\sqrt {x^2+y(x)^2}\right ) y'(x)=y(x)\)
   4.15.34 \(x \left (1-\sqrt {x^2-y(x)^2}\right ) y'(x)=y(x)\)
   4.15.35 \(x \left (\sqrt {x^2+y(x)^2}+x\right ) y'(x)+\sqrt {x^2+y(x)^2} y(x)=0\)
   4.15.36 \(x y(x) \left (\sqrt {x^2-y(x)^2}+x\right ) y'(x)=x y(x)^2-\left (x^2-y(x)^2\right )^{3/2}\)
   4.15.37 \(\left (x-y(x)^2 \sqrt {y(x)^2-x^2}\right ) y'(x)=y(x) \left (x \sqrt {y(x)^2-x^2}+1\right )\)
   4.15.38 \(\left (x \sqrt {x^2+y(x)^2+1}-y(x) \left (x^2+y(x)^2\right )\right ) y'(x)=\sqrt {x^2+y(x)^2+1} y(x)+x \left (x^2+y(x)^2\right )\)
   4.15.39 \(y'(x) \left (x^2 \sec (x) \cos (y(x))+1\right )-y(x) \tan (x)+\sec (x) \sin (y(x))=0\)
   4.15.40 \(y'(x) \cos (y(x)) (\cos (y(x))-\sin (A) \sin (x))+\cos (x) (\cos (x)-\sin (A) \sin (y(x)))=0\)
   4.15.41 \(y'(x) (a \cos (a y(x)+b x)-b \sin (a x+b y(x)))-a \sin (a x+b y(x))+b \cos (a y(x)+b x)=0\)
   4.15.42 \(y'(x) (a \sin (x y(x))-\sin (y(x))+\cos (y(x)+x))+y(x) \sin (x y(x))+\cos (y(x)+x)+\cos (x)=0\)
   4.15.43 \(y'(x) (\cos (x) \sec (y(x))+x)+\tan (y(x))-y(x) \sin (x) \sec (y(x))=0\)
   4.15.44 \(y'(x) \left (x^2+2 y(x) \sin (x) \sec (y(x))\right )+2 x \tan (y(x))+y(x)^2 \cos (x) \sec (y(x))=0\)
   4.15.45 \(y'(x) \left (x^2 \sec ^2(y(x))-6 x y(x)+2\right )-3 y(x)^2+2 x \tan (y(x))=0\)
   4.15.46 \(y'(x) ((y(x)+x) \tan (y(x))+1)+1=0\)
   4.15.47 \(x y'(x) \left (x-y(x) \tan \left (\frac {y(x)}{x}\right )\right )+y(x) \left (y(x) \tan \left (\frac {y(x)}{x}\right )+x\right )=0\)
   4.15.48 \(\left (x e^{y(x)}+e^x\right ) y'(x)+e^x y(x)+e^{y(x)}=0\)
   4.15.49 \(y'(x) (-\log (y(x))-2 x+1)+2 y(x)=0\)
   4.15.50 \(y'(x) (x \cosh (y(x))+\sinh (x))+\sinh (y(x))+y(x) \cosh (x)=0\)