Internal problem ID [10634]
Internal file name [OUTPUT/9582_Monday_June_06_2022_03_10_56_PM_41579351/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_Riccati]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-y^{2}=-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati Special trying Riccati sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (diff(diff(f(x), x), x))*y(x)/f(x), y(x)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 44
dsolve(diff(y(x),x)=y(x)^2-diff(f(x),x$2)/f(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\left (\int \frac {1}{f \left (x \right )^{2}}d x \right ) \left (\frac {d}{d x}f \left (x \right )\right ) f \left (x \right )-\left (\frac {d}{d x}f \left (x \right )\right ) c_{1} f \left (x \right )-1}{\left (\int \frac {1}{f \left (x \right )^{2}}d x +c_{1} \right ) f \left (x \right )^{2}} \]
✓ Solution by Mathematica
Time used: 0.365 (sec). Leaf size: 132
DSolve[y'[x]==y[x]^2-f''[x]/f[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{\left (f(x) K[2]+f'(x)\right )^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2-f''(K[1])\right )}{\left (f(K[1]) K[2]+f'(K[1])\right )^3}-\frac {2 K[2]}{\left (f(K[1]) K[2]+f'(K[1])\right )^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2-f''(K[1])}{f(K[1]) \left (f(K[1]) y(x)+f'(K[1])\right )^2}dK[1]=c_1,y(x)\right ] \]