Internal problem ID [9892]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-2. Equations containing arbitrary
functions and their derivatives.
Problem number: 42.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}+\frac {f^{\prime \prime }\relax (x )}{f \relax (x )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 55
dsolve(diff(y(x),x)=y(x)^2-diff(f(x),x$2)/f(x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\int \frac {1}{f \relax (x )^{2}}d x \right ) \left (\frac {d}{d x}f \relax (x )\right )+\left (\frac {d}{d x}f \relax (x )\right ) c_{1}}{\left (\int \frac {1}{f \relax (x )^{2}}d x +c_{1}\right ) f \relax (x )}-\frac {1}{\left (\int \frac {1}{f \relax (x )^{2}}d x +c_{1}\right ) f \relax (x )^{2}} \]
✓ Solution by Mathematica
Time used: 0.198 (sec). Leaf size: 132
DSolve[y'[x]==y[x]^2-f''[x]/f[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{\left (f(x) K[2]+f'(x)\right )^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2-f''(K[1])\right )}{\left (f(K[1]) K[2]+f'(K[1])\right )^3}-\frac {2 K[2]}{\left (f(K[1]) K[2]+f'(K[1])\right )^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2-f''(K[1])}{f(K[1]) \left (f(K[1]) y(x)+f'(K[1])\right )^2}dK[1]=c_1,y(x)\right ] \]