4.42 Problems 2051 to 2100

   4.42.1 \(y'(x) y''(x)=x^2 y(x) y'(x)+x y(x)^2\)
   4.42.2 \(\left (2 x^2 y'(x)+y(x)^2\right ) y''(x)+2 (y(x)+x) y'(x)^2+x y'(x)+y(x)=0\)
   4.42.3 \(a y(x)^2+x^3 y'(x) y''(x)=0\)
   4.42.4 \(\text {f1} y'(x) y''(x)+\text {f2} y(x) y''(x)+\text {f3} y'(x)^2+\text {f4} y(x) y'(x)+\text {f5} y(x)^2=0\)
   4.42.5 \(3 y(x) y'(x) y''(x)=y'(x)^3-1\)
   4.42.6 \(\left (x^2+2 y(x)^2 y'(x)\right ) y''(x)+2 y(x) y'(x)^3+3 x y'(x)+y(x)=0\)
   4.42.7 \(\left (x-y'(x)^2\right ) y''(x)=x^2-y'(x)\)
   4.42.8 \(\left (y'(x)^2+y(x)^2\right ) y''(x)+y(x)^3=0\)
   4.42.9 \(y''(x) \left (a \left (x y'(x)-y(x)\right )+y'(x)^2\right )=b\)
   4.42.10 \(4 y(x) y'(x)^2 y''(x)=y'(x)^4+3\)
   4.42.11 \(y''(x) f\left (y'(x)\right )+g(y(x)) y'(x)+h(x)=0\)
   4.42.12 \(y''(x)^2=a+b y(x)\)
   4.42.13 \(y''(x)^2=a+b y'(x)^2\)
   4.42.14 \(y''(x)^2-x y''(x)+y'(x)=0\)
   4.42.15 \(a^2 y''(x)^2=\left (y'(x)^2+1\right )^3\)
   4.42.16 \(a x+x y''(x)^2-2 y'(x) y''(x)=0\)
   4.42.17 \(\left (x y''(x)-y'(x)\right )^2=y''(x)^2+1\)
   4.42.18 \(2 \left (x^2+1\right ) y''(x)^2+2 \left (x-y'(x)\right ) y'(x)-x \left (4 y'(x)+x\right ) y''(x)=2 y(x)\)
   4.42.19 \(3 x^2 y''(x)^2+4 y'(x)^2-2 \left (3 x y'(x)+y(x)\right ) y''(x)=0\)
   4.42.20 \((2-9 x) x^2 y''(x)^2+6 y(x) y''(x)-6 (1-6 x) x y'(x) y''(x)=36 x y'(x)^2\)
   4.42.21 \(\text {f0} y''(x)^2+\text {f1} y'(x) y''(x)+\text {f2} y(x) y''(x)+\text {g0} y'(x)^2+\text {g1} y(x) y'(x)+h y(x)^2=0\)
   4.42.22 \(y(x) y''(x)+4 y(x) y'(x)^3-y'(x)^2=0\)
   4.42.23 \(\left (y(x) y''(x)+y'(x)^2+1\right )^2=\left (y'(x)^2+1\right )^3\)
   4.42.24 \(y''(x)^2 \left (a^2-b^2 y(x)^2\right )+y'(x)^2 \left (1-b^2 y'(x)^2\right )+2 b^2 y(x) y'(x)^2 y''(x)=0\)
   4.42.25 \(\left (x^2 y(x) y''(x)+x^2 \left (-y'(x)^2\right )+y(x)^2\right )^2=4 x y(x) \left (x y'(x)-y(x)\right )^3\)
   4.42.26 \(y''(x)^3=12 y'(x) \left (x y''(x)-2 y'(x)\right )\)
   4.42.27 \(32 y''(x) \left (x y''(x)-y'(x)\right )^3+\left (2 y(x) y''(x)-y'(x)^2\right )^3=0\)
   4.42.28 \(f\left (y''(x)\right )+x y''(x)=y'(x)\)
   4.42.29 \(y'(x) f\left (\frac {y''(x)}{y'(x)}\right )=y'(x)^2-y(x) y''(x)\)
   4.42.30 \(f\left (y''(x),y'(x)-x y''(x),\frac {1}{2} x^2 y''(x)-x y'(x)+y(x)\right )=0\)
   4.42.31 \(f\left (x,y''(x)\right )=0\)
   4.42.32 \(f\left (y(x),y''(x)\right )=0\)
   4.42.33 \(f\left (y'(x),y''(x)\right )=0\)
   4.42.34 \(f\left (x,y'(x),y''(x)\right )=0\)
   4.42.35 \(f\left (y(x),y'(x),y''(x)\right )=0\)
   4.42.36 \(y'''(x)=0\)
   4.42.37 \(y'''(x)=\cos (x)+1\)
   4.42.38 \(y'''(x)+\sin (x)=0\)
   4.42.39 \(y'''(x)=\sin ^3(x)\)
   4.42.40 \(y'''(x)=y(x)\)
   4.42.41 \(y'''(x)=x^2+y(x)\)
   4.42.42 \(y'''(x)=y(x)+e^x x+\cos ^2(x)\)
   4.42.43 \(y'''(x)+a y(x)=0\)
   4.42.44 \(y'''(x)=x y(x)\)
   4.42.45 \(y'''(x)+y'(x)=0\)
   4.42.46 \(y'''(x)=y'(x)\)
   4.42.47 \(y'''(x)+y'(x)=x^3+\cos (x)\)
   4.42.48 \(y'''(x)-2 y'(x)+4 y(x)=0\)
   4.42.49 \(y'''(x)-2 y'(x)+4 y(x)=e^x \cos (x)\)
   4.42.50 \(y'''(x)-3 y'(x)+2 y(x)=0\)