4.41 Problems 2001 to 2050

   4.41.1 \(8 \left (1-x^3\right ) y(x) y''(x)+4 \left (1-x^3\right ) y'(x)^2-12 x^2 y(x) y'(x)+3 x y(x)^2=0\)
   4.41.2 \(\sqrt {a^2+x^2} \left (b y'(x)^2+y(x) y''(x)\right )=y(x) y'(x)\)
   4.41.3 \(\sqrt {a^2-x^2} \left (x y(x) y''(x)-x y'(x)^2-y(x) y'(x)\right )=b x y'(x)^2\)
   4.41.4 \(\text {f0}(x) y(x) y''(x)+\text {f1}(x) y'(x)^2+\text {f2}(x) y(x) y'(x)+\text {f3}(x) y(x)^2=0\)
   4.41.5 \(4 f(x) y(x) y''(x)=y'(x) \left (2 f'(x)-6 f(x) y(x)^2\right )+2 y(x)^3 f'(x)+3 f(x) g(x) y(x)^2+3 f(x) y'(x)^2-f(x) y(x)^4+4 f(x)^2 y(x)\)
   4.41.6 \(y(x)^2 y''(x)=a\)
   4.41.7 \(a x+y(x)^2 y''(x)+y(x) y'(x)^2=0\)
   4.41.8 \(y(x)^2 y''(x)+y(x) y'(x)^2=a+b x\)
   4.41.9 \(\left (y(x)^2+1\right ) y''(x)+(1-2 y(x)) y'(x)^2=0\)
   4.41.10 \(\left (y(x)^2+1\right ) y''(x)=3 y(x) y'(x)^2\)
   4.41.11 \(\left (y(x)^2+1\right ) y''(x)=(a+3 y(x)) y'(x)^2\)
   4.41.12 \(\left (y(x)^2+1\right ) y''(x)+y'(x) \left (y'(x)^2+1\right )=0\)
   4.41.13 \(\left (y(x)^2+x\right ) y''(x)+2 y(x) y'(x)^2+2 y'(x)=a\)
   4.41.14 \(\left (y(x)^2+x\right ) y''(x)=2 \left (x-y(x)^2\right ) y'(x)^3-y'(x) \left (4 y(x) y'(x)+1\right )\)
   4.41.15 \(\left (x^2+y(x)^2\right ) y''(x)=\left (y(x)^2+1\right ) \left (x y'(x)-y(x)\right )\)
   4.41.16 \(\left (x^2+y(x)^2\right ) y''(x)=2 \left (y(x)^2+1\right ) \left (x y'(x)-y(x)\right )\)
   4.41.17 \(2 (1-y(x)) y(x) y''(x)=(1-2 y(x)) y'(x)^2\)
   4.41.18 \(2 (1-y(x)) y(x) y''(x)=f(x) (1-y(x)) y(x) y'(x)+(1-2 y(x)) y'(x)^2\)
   4.41.19 \(2 (1-y(x)) y(x) y''(x)=(1-3 y(x)) y'(x)^2\)
   4.41.20 \(2 (1-y(x)) y(x) y''(x)=4 y(x) y'(x) (f(x)+g(x) y(x))+(1-3 y(x)) y'(x)^2\)
   4.41.21 \(2 (1-y(x)) y(x) y''(x)=-4 y(x)^2 (1-y(x)) \left (f'(x)+f(x)^2+g'(x)-g(x)^2\right )-4 y(x) y'(x) (f(x)+g(x) y(x))+(1-y(x))^3 \left (-\left (\text {F0}(x)^2-\text {G0}(x)^2 y(x)^2\right )\right )+(1-3 y(x)) y'(x)^2\)
   4.41.22 \(3 (1-y(x)) y(x) y''(x)=2 (1-2 y(x)) y'(x)^2\)
   4.41.23 \(4 (1-y(x)) y(x) y''(x)=3 (1-2 y(x)) y'(x)^2\)
   4.41.24 \(x y(x)^2 y''(x)=a\)
   4.41.25 \(x y(x)^2 y''(x)=\left (a-y(x)^2\right ) y'(x)+x y(x) y'(x)^2\)
   4.41.26 \(x^2 y(x)^2 y''(x)=\left (x^2+y(x)^2\right ) \left (x y'(x)-y(x)\right )\)
   4.41.27 \(\left (a^2-x^2\right ) \left (a^2-y(x)^2\right ) y''(x)+\left (a^2-x^2\right ) y(x) y'(x)^2=x \left (a^2-y(x)^2\right ) y'(x)\)
   4.41.28 \((1-y(x))^3 \left (\text {a0}+\text {a1} y(x)^2\right )+\text {a2} x (1-y(x)) y(x)^2+\text {a3} x^3 y(x)^2 (y(x)+1)+2 x^2 (1-y(x)) y(x) y''(x)-x^2 (1-3 y(x)) y'(x)^2+2 x (1-y(x)) y(x) y'(x)=0\)
   4.41.29 \(x^3 y(x)^2 y''(x)+(y(x)+x) \left (x y'(x)-y(x)\right )^3=0\)
   4.41.30 \(y(x)^3 y''(x)=a^2\)
   4.41.31 \(y(x) \left (y(x)^2+1\right ) y''(x)+\left (1-3 y(x)^2\right ) y'(x)^2=0\)
   4.41.32 \(2 y(x)^3 y''(x)+y(x)^2 y'(x)^2=2\)
   4.41.33 \(2 (1-y(x)) y(x) (a-y(x)) y''(x)+(-(1-y(x)) (a-y(x))+y(x) (a-y(x))+(1-y(x)) y(x)) y'(x)^2=\text {a0} y(x)^2 \left (1-y(x)^2\right ) (a-y(x))^2+\text {a2} y(x)^2 (a-y(x))^2+\text {a3} (1-y(x))^2 (a-y(x))^2+\text {a1} (1-y(x))^2 y(x)^2\)
   4.41.34 \(2 (a-y(x)) (b-y(x)) (c-y(x)) y''(x)+y'(x)^2 ((a-y(x)) (b-y(x))+(a-y(x)) (c-y(x))+(b-y(x)) (c-y(x)))=\text {a0} (a-y(x))^2 (b-y(x))^2 (c-y(x))^2+2 \text {a2} (a-y(x))^2 (c-y(x))^2+\text {a3} (a-y(x))^2 (b-y(x))^2+\text {a1} (b-y(x))^2 (c-y(x))^2\)
   4.41.35 \(2 (1-x) x (1-y(x)) (x-y(x)) y(x) y''(x)=2 (1-y(x)) \left (x^2-2 x y(x)+y(x)\right ) y(x) y'(x)+(1-x) x \left (3 y(x)^2-2 x y(x)-2 y(x)+x\right ) y'(x)^2-\left (1-y(x)^2\right ) y(x)^2\)
   4.41.36 \(2 (1-x) x (1-y(x)) (x-y(x)) y(x) y''(x)=f(x) ((1-y(x)) (x-y(x)) y(x))^{3/2}+2 (1-y(x)) \left (x^2-2 x y(x)+y(x)\right ) y(x) y'(x)+(1-x) x \left (3 y(x)^2-2 x y(x)-2 y(x)+x\right ) y'(x)^2-\left (1-y(x)^2\right ) y(x)^2\)
   4.41.37 \(2 (1-x)^2 x^2 (1-y(x)) (x-y(x)) y(x) y''(x)=\text {a0} x (x-y(x))^2 (1-y(x))^2+\text {a1} (1-x) (x-y(x))^2 y(x)^2+(\text {a2}-1) (1-x) x y(x)^2 (1-y(x))^2+\text {a3} (x-y(x))^2 y(x)^2 (1-y(x))^2+2 (1-x) x y(x) \left (x^2-2 x y(x)+y(x)\right ) (1-y(x))^2 y'(x)+(1-x)^2 x^2 \left (3 y(x)^2-2 x y(x)-2 y(x)+x\right ) y'(x)^2\)
   4.41.38 \(b \sqrt {\left (1-y(x)^2\right ) \left (1-a^2 y(x)^2\right )} y'(x)^2+\left (1-y(x)^2\right ) \left (1-a^2 y(x)^2\right ) y''(x)+y(x) \left (-2 a^2 y(x)^2+a^2+1\right )=0\)
   4.41.39 \(a^2 y(x)+\left (x^2+y(x)^2\right )^2 y''(x)=0\)
   4.41.40 \(y''(x) \left (a+2 b x+c x^2+y(x)^2\right )^2+A y(x)=0\)
   4.41.41 \(\text {f0}(y(x)) y''(x)+\text {f1}(y(x)) y'(x)^2+\text {f2}(y(x)) y'(x)+\text {f3}(y(x))=0\)
   4.41.42 \(\sqrt {y(x)} y''(x)=a\)
   4.41.43 \(\sqrt {y(x)} y''(x)=2 (a+b x)\)
   4.41.44 \(y''(x) X(x,y(x))^3=1\)
   4.41.45 \(y''(x) \left (\text {a0}+\text {a1} \sin ^2(y(x))\right )+\text {a2} y(x) \left (\text {a1} \sin ^2(y(x))+\text {a3}\right )+\text {a1} y'(x)^2+\text {a1} y'(x)^2 \sin (y(x)) \cos (y(x))=0\)
   4.41.46 \(y(x) y''(x) (1-\log (y(x)))+y'(x)^2 (\log (y(x))+1)=0\)
   4.41.47 \(f(y(x)) y''(x)=y'(x)^2 f'(y(x))-g(x) f(y(x)) y'(x)-h(x) f(y(x))^2\)
   4.41.48 \(f(y(x)) y''(x)=f(y(x))^2 \text {F0}\left (x,\frac {y'(x)}{f(y(x))}\right )+y'(x)^2 f'(y(x))\)
   4.41.49 \(a y'(x)^2 f'(y(x))+f(y(x)) y''(x)+g(y(x))=0\)
   4.41.50 \(y'(x) y''(x)=a^2 x\)