Internal problem ID [10618]
Internal file name [OUTPUT/9566_Monday_June_06_2022_03_09_41_PM_20497896/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 1.2.8-1. Equations containing
arbitrary functions (but not containing their derivatives).
Problem number: 26.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\[ \boxed {x y^{\prime }-f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}=-a} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {a^{2} f \left (x \right ) \ln \left (x \right )^{2}+2 f \left (x \right ) \ln \left (x \right ) a y +f \left (x \right ) y^{2}-a}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {a^{2} f \left (x \right ) \ln \left (x \right )^{2}}{x}+\frac {2 f \left (x \right ) \ln \left (x \right ) a y}{x}+\frac {f \left (x \right ) y^{2}}{x}-\frac {a}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {-a +a^{2} f \left (x \right ) \ln \left (x \right )^{2}}{x}\), \(f_1(x)=\frac {2 a f \left (x \right ) \ln \left (x \right )}{x}\) and \(f_2(x)=\frac {f \left (x \right )}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {f \left (x \right ) u}{x}} \tag {1} \end {align*}
Using the above substitution in the given ODE results (after some simplification)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=-\frac {f \left (x \right )}{x^{2}}+\frac {f^{\prime }\left (x \right )}{x}\\ f_1 f_2 &=\frac {2 a f \left (x \right )^{2} \ln \left (x \right )}{x^{2}}\\ f_2^2 f_0 &=\frac {f \left (x \right )^{2} \left (-a +a^{2} f \left (x \right ) \ln \left (x \right )^{2}\right )}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {f \left (x \right ) u^{\prime \prime }\left (x \right )}{x}-\left (-\frac {f \left (x \right )}{x^{2}}+\frac {f^{\prime }\left (x \right )}{x}+\frac {2 a f \left (x \right )^{2} \ln \left (x \right )}{x^{2}}\right ) u^{\prime }\left (x \right )+\frac {f \left (x \right )^{2} \left (-a +a^{2} f \left (x \right ) \ln \left (x \right )^{2}\right ) u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (-\frac {f \left (x \right )}{x^{2}}+\frac {f^{\prime }\left (x \right )}{x}+\frac {2 a f \left (x \right )^{2} \ln \left (x \right )}{x^{2}}\right ) x \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )}+\frac {f \left (x \right ) \left (-a +a^{2} f \left (x \right ) \ln \left (x \right )^{2}\right ) \textit {\_Y} \left (x \right )}{x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {d}{d x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (-\frac {f \left (x \right )}{x^{2}}+\frac {f^{\prime }\left (x \right )}{x}+\frac {2 a f \left (x \right )^{2} \ln \left (x \right )}{x^{2}}\right ) x \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )}+\frac {f \left (x \right ) \left (-a +a^{2} f \left (x \right ) \ln \left (x \right )^{2}\right ) \textit {\_Y} \left (x \right )}{x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (-\frac {f \left (x \right )}{x^{2}}+\frac {f^{\prime }\left (x \right )}{x}+\frac {2 a f \left (x \right )^{2} \ln \left (x \right )}{x^{2}}\right ) x \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )}+\frac {f \left (x \right ) \left (-a +a^{2} f \left (x \right ) \ln \left (x \right )^{2}\right ) \textit {\_Y} \left (x \right )}{x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x}{f \left (x \right ) \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (-\frac {f \left (x \right )}{x^{2}}+\frac {f^{\prime }\left (x \right )}{x}+\frac {2 a f \left (x \right )^{2} \ln \left (x \right )}{x^{2}}\right ) x \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )}+\frac {f \left (x \right ) \left (-a +a^{2} f \left (x \right ) \ln \left (x \right )^{2}\right ) \textit {\_Y} \left (x \right )}{x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x^{2} f \left (x \right )-2 \left (f \left (x \right )^{2} \ln \left (x \right ) a +\frac {x f^{\prime }\left (x \right )}{2}-\frac {f \left (x \right )}{2}\right ) x \textit {\_Y}^{\prime }\left (x \right )+f \left (x \right )^{2} a \left (f \left (x \right ) \ln \left (x \right )^{2} a -1\right ) \textit {\_Y} \left (x \right )}{x^{2} f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x}{f \left (x \right ) \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x^{2} f \left (x \right )-2 \left (f \left (x \right )^{2} \ln \left (x \right ) a +\frac {x f^{\prime }\left (x \right )}{2}-\frac {f \left (x \right )}{2}\right ) x \textit {\_Y}^{\prime }\left (x \right )+f \left (x \right )^{2} a \left (f \left (x \right ) \ln \left (x \right )^{2} a -1\right ) \textit {\_Y} \left (x \right )}{x^{2} f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x^{2} f \left (x \right )-2 \left (f \left (x \right )^{2} \ln \left (x \right ) a +\frac {x f^{\prime }\left (x \right )}{2}-\frac {f \left (x \right )}{2}\right ) x \textit {\_Y}^{\prime }\left (x \right )+f \left (x \right )^{2} a \left (f \left (x \right ) \ln \left (x \right )^{2} a -1\right ) \textit {\_Y} \left (x \right )}{x^{2} f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x}{f \left (x \right ) \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x^{2} f \left (x \right )-2 \left (f \left (x \right )^{2} \ln \left (x \right ) a +\frac {x f^{\prime }\left (x \right )}{2}-\frac {f \left (x \right )}{2}\right ) x \textit {\_Y}^{\prime }\left (x \right )+f \left (x \right )^{2} a \left (f \left (x \right ) \ln \left (x \right )^{2} a -1\right ) \textit {\_Y} \left (x \right )}{x^{2} f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x^{2} f \left (x \right )-2 \left (f \left (x \right )^{2} \ln \left (x \right ) a +\frac {x f^{\prime }\left (x \right )}{2}-\frac {f \left (x \right )}{2}\right ) x \textit {\_Y}^{\prime }\left (x \right )+f \left (x \right )^{2} a \left (f \left (x \right ) \ln \left (x \right )^{2} a -1\right ) \textit {\_Y} \left (x \right )}{x^{2} f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x}{f \left (x \right ) \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x^{2} f \left (x \right )-2 \left (f \left (x \right )^{2} \ln \left (x \right ) a +\frac {x f^{\prime }\left (x \right )}{2}-\frac {f \left (x \right )}{2}\right ) x \textit {\_Y}^{\prime }\left (x \right )+f \left (x \right )^{2} a \left (f \left (x \right ) \ln \left (x \right )^{2} a -1\right ) \textit {\_Y} \left (x \right )}{x^{2} f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}=-a \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}-a}{x} \end {array} \]
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 sub-methods: <- Riccati particular case Kamke (d) successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 24
dsolve(x*diff(y(x),x)=f(x)*(y(x)+a*ln(x))^2-a,y(x), singsol=all)
\[ y \left (x \right ) = -a \ln \left (x \right )+\frac {1}{c_{1} -\left (\int \frac {f \left (x \right )}{x}d x \right )} \]
✓ Solution by Mathematica
Time used: 0.48 (sec). Leaf size: 42
DSolve[x*y'[x]==f[x]*(y[x]+a*Log[x])^2-a,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -a \log (x)+\frac {1}{-\int _1^x\frac {f(K[2])}{K[2]}dK[2]+c_1} \\ y(x)\to -a \log (x) \\ \end{align*}