4.38.1 \(a y'(x)+b x e^{y(x)}+x y''(x)=0\)
4.38.2 \(\left (2-a x^2\right ) y'(x)+x y''(x)=0\)
4.38.3 \(x y''(x)=(1-y(x)) y'(x)\)
4.38.4 \(x y''(x)+x y'(x)^2=y'(x)\)
4.38.5 \(x y''(x)=x y'(x)^2+y'(x)\)
4.38.6 \(x y''(x)+2 x y'(x)^2-2 y'(x)=0\)
4.38.7 \(x y''(x)=x^2 y'(x)^2-2 y'(x)-y(x)^2\)
4.38.8 \(a x^2 y'(x)^2+x y''(x)+2 y'(x)=b\)
4.38.9 \(\left (a x y'(x)-y(x)\right )^2+x y''(x)=b\)
4.38.10 \(x y''(x)=y'(x)^3+y'(x)\)
4.38.11 \(x y''(x)+2 y'(x)=a x^{2 k} y'(x)^k\)
4.38.12 \(2 x y''(x)+y'(x)^3+y'(x)=0\)
4.38.13 \(a y(x) \left (1-y(x)^n\right )+x^2 y''(x)=0\)
4.38.14 \(a e^{y(x)-1}+x^2 y''(x)=0\)
4.38.15 \((a+1) x y'(x)+x^2 y''(x)=x^k f\left (x^k y(x),k y(x)+x y'(x)\right )\)
4.38.16 \(x^2 y''(x)+y'(x)^2=0\)
4.38.17 \(x^2 y''(x)=\left (3 x-2 y'(x)\right ) y'(x)\)
4.38.18 \(x^2 y''(x)+x^2 y'(x)^2+4 x y'(x)+2=0\)
4.38.19 \(x^2 y''(x)=x^4 y'(x)^2-4 x^2 y(x)^2+6 y(x)\)
4.38.20 \(a \left (x y'(x)-y(x)\right )^2+x^2 y''(x)=b x^2\)
4.38.21 \(a x^4 y'(x)^2+x^2 y''(x)+2 x y(x)=b\)
4.38.22 \(a y(x) y'(x)^2+b x+x^2 y''(x)=0\)
4.38.23 \(x^2 y''(x)=\sqrt {a x^2 y'(x)^2+b y(x)^2}\)
4.38.24 \(x^2 y''(x)=y(x) f\left (\genfrac {}{}{}{}{x y'(x)}{y(x)}\right )\)
4.38.25 \(\left (x^2+1\right ) y''(x)+y'(x)^2+1=0\)
4.38.26 \(a y(x)^3+9 x^2 y''(x)+2 y(x)=0\)
4.38.27 \(x^3 y''(x)-x^2 y'(x)=3-x^2\)
4.38.28 \(x^3 \left (y''(x)+y(x) y'(x)-y(x)^3\right )+12 x y(x)+24=0\)
4.38.29 \(x^3 y''(x)=a \left (x y'(x)-y(x)\right )^2\)
4.38.30 \(2 x^3 y''(x)+x^2 (2 x y(x)+9) y'(x)+x y(x) \left (-2 x^2 y(x)^2+3 x y(x)+12\right )-6=0\)
4.38.31 \(x^4 y''(x)=x \left (x^2+2 y(x)\right ) y'(x)-4 y(x)^2\)
4.38.32 \(x^4 y''(x)=x^2 y'(x) \left (y'(x)+x\right )-4 y(x)^2\)
4.38.33 \(x^4 y''(x)+\left (x y'(x)-y(x)\right )^3=0\)
4.38.34 \(x^a y''(x)+y(x)^b=0\)
4.38.35 \(\left (1-12 x^2\right ) \left (3 y'(x)+y(x)^2\right )+2 x \left (1-4 x^2\right ) \left (y''(x)+y(x) y'(x)-y(x)^3\right )-48 x y(x)+24=0\)
4.38.36 \(a x y(x)+b-\left (k x^{k-1}-12 x^2\right ) \left (3 y'(x)+y(x)^2\right )+2 \left (x^k-4 x^3\right ) \left (y''(x)+y(x) y'(x)-y(x)^3\right )=0\)
4.38.37 \(\sqrt {x} y''(x)=y(x)^{3/2}\)
4.38.38 \(x^{3/2} y''(x)=f\left (\genfrac {}{}{}{}{y(x)}{\sqrt {x}}\right )\)
4.38.39 \(y''(x) \left (a+2 b x+c x^2\right )^{3/2}=f\left (\genfrac {}{}{}{}{x}{\sqrt {a+2 b x+c x^2}}\right )\)
4.38.40 \(f(x) f'(x) y'(x)+f(x)^2 y''(x)=g\left (y(x),f(x) y'(x)\right )\)
4.38.41 \(f(x)^2 y''(x)=y'(x) \left (3 f(x) f'(x)-f(x)^2 y(x)+3 f(x)^3\right )-24 f(x)^4\)
4.38.42 \(f(x)^2 y''(x)=-a f(x)^5+3 f(x) f'(x)-f(x)^2 y(x)+3 f(x)^3\)
4.38.43 \(2 f(x)^2 y''(x)=f(x) y'(x) \left (3 f'(x)-2 f(x) y(x)\right )+f(x) y(x)^2 f'(x)+y(x) \left (f(x) f''(x)-2 f'(x)^2-2 f(x)^3\right )+2 f(x)^2 y(x)^3\)
4.38.44 \(y(x) y''(x)=a\)
4.38.45 \(y(x) y''(x)=y'(x)^2\)
4.38.46 \(y(x) y''(x)+y'(x)^2=0\)
4.38.47 \(y(x) y''(x)=y'(x)^2-a^2\)
4.38.48 \(y(x) y''(x)+y'(x)^2=a^2\)
4.38.49 \(y(x) y''(x)+y'(x)^2+y(x)^2=0\)
4.38.50 \(2 a^2 y(x)^2+y(x) y''(x)+y'(x)^2=0\)