61.20.9 problem 42

Internal problem ID [12311]
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
Date solved : Tuesday, January 28, 2025 at 02:34:51 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 44

dsolve(diff(y(x),x)=y(x)^2-diff(f(x),x$2)/f(x),y(x), singsol=all)
 

\[ y = \frac {-\left (\int \frac {1}{f^{2}}d x \right ) f^{\prime } f-f^{\prime } c_{1} f-1}{\left (\int \frac {1}{f^{2}}d x +c_{1} \right ) f^{2}} \]

✓ Solution by Mathematica

Time used: 0.238 (sec). Leaf size: 132

DSolve[D[y[x],x]==y[x]^2-D[ f[x],{x,2}]/f[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{\left (f(x) K[2]+f''(x)\right )^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2-f''''(K[1])\right )}{\left (f(K[1]) K[2]+f''(K[1])\right )^3}-\frac {2 K[2]}{\left (f(K[1]) K[2]+f''(K[1])\right )^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2-f''''(K[1])}{f(K[1]) \left (f(K[1]) y(x)+f''(K[1])\right )^2}dK[1]=c_1,y(x)\right ] \]