Internal problem ID [9019]
Internal file name [OUTPUT/7954_Monday_June_06_2022_01_00_44_AM_10061394/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 685.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati", "homogeneousTypeD2"
Maple gives the following as the ode type
[[_homogeneous, `class D`], _Riccati]
\[ \boxed {y^{\prime }-\frac {y+\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x^{3}+7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x y^{2}}{x}=0} \]
Using the change of variables \(y = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} 7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x^{3} u \left (x \right )^{2}+\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x^{3}-\left (u^{\prime }\left (x \right ) x +u \left (x \right )\right ) x +u \left (x \right ) x = 0 \end {align*}
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x \left (7 u^{2}+1\right ) \end {align*}
Where \(f(x)=\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x\) and \(g(u)=7 u^{2}+1\). Integrating both sides gives \begin{align*} \frac {1}{7 u^{2}+1} \,du &= \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x \,d x \\ \int { \frac {1}{7 u^{2}+1} \,du} &= \int {\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x \,d x} \\ \frac {\sqrt {7}\, \arctan \left (u \sqrt {7}\right )}{7}&=\frac {\ln \left (\left (x -1\right ) \left (x +1\right )\right ) \left (x -1\right ) \left (x +1\right )}{2}-\frac {\left (x -1\right ) \left (x +1\right )}{2}+c_{2} \\ \end{align*} The solution is \[ \frac {\sqrt {7}\, \arctan \left (u \left (x \right ) \sqrt {7}\right )}{7}-\frac {\ln \left (\left (x -1\right ) \left (x +1\right )\right ) \left (x -1\right ) \left (x +1\right )}{2}+\frac {\left (x -1\right ) \left (x +1\right )}{2}-c_{2} = 0 \] Replacing \(u(x)\) in the above solution by \(\frac {y}{x}\) results in the solution for \(y\) in implicit form \begin {align*} \frac {\sqrt {7}\, \arctan \left (\frac {y \sqrt {7}}{x}\right )}{7}-\frac {\ln \left (\left (x -1\right ) \left (x +1\right )\right ) \left (x -1\right ) \left (x +1\right )}{2}+\frac {\left (x -1\right ) \left (x +1\right )}{2}-c_{2} = 0\\ \frac {\sqrt {7}\, \arctan \left (\frac {y \sqrt {7}}{x}\right )}{7}-\frac {x^{2} \ln \left (x^{2}-1\right )}{2}+\frac {\ln \left (x^{2}-1\right )}{2}+\frac {x^{2}}{2}-\frac {1}{2}-c_{2} = 0 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \frac {\sqrt {7}\, \arctan \left (\frac {y \sqrt {7}}{x}\right )}{7}-\frac {x^{2} \ln \left (x^{2}-1\right )}{2}+\frac {\ln \left (x^{2}-1\right )}{2}+\frac {x^{2}}{2}-\frac {1}{2}-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ \frac {\sqrt {7}\, \arctan \left (\frac {y \sqrt {7}}{x}\right )}{7}-\frac {x^{2} \ln \left (x^{2}-1\right )}{2}+\frac {\ln \left (x^{2}-1\right )}{2}+\frac {x^{2}}{2}-\frac {1}{2}-c_{2} = 0 \] Verified OK.
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y +\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x^{3}+7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x \,y^{2}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = x^{2} \ln \left (\left (x -1\right ) \left (x +1\right )\right )+7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) y^{2}+\frac {y}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=x^{2} \ln \left (\left (x -1\right ) \left (x +1\right )\right )\), \(f_1(x)=\frac {1}{x}\) and \(f_2(x)=7 \ln \left (\left (x -1\right ) \left (x +1\right )\right )\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) u} \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 {14 x}{\left (x +1\right ) \left (x -1\right )}\\ f_1 f_2 &=\frac {7 \ln \left (\left (x -1\right ) \left (x +1\right )\right )}{x}\\ f_2^2 f_0 &=49 \ln \left (\left (x -1\right ) \left (x +1\right )\right )^{3} x^{2} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} 7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) u^{\prime \prime }\left (x \right )-\left (\frac {14 x}{\left (x +1\right ) \left (x -1\right )}+\frac {7 \ln \left (\left (x -1\right ) \left (x +1\right )\right )}{x}\right ) u^{\prime }\left (x \right )+49 \ln \left (\left (x -1\right ) \left (x +1\right )\right )^{3} x^{2} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \sin \left (\frac {\sqrt {7}\, \left (x -1\right ) \left (x +1\right ) \left (\ln \left (x^{2}-1\right )-1\right )}{2}\right )+c_{2} \cos \left (\frac {\sqrt {7}\, \left (x -1\right ) \left (x +1\right ) \left (\ln \left (x^{2}-1\right )-1\right )}{2}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \sqrt {7}\, \ln \left (x^{2}-1\right ) x \left (c_{1} \cos \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )-c_{2} \sin \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\sqrt {7}\, \ln \left (x^{2}-1\right ) x \left (c_{1} \cos \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )-c_{2} \sin \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )\right )}{7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) \left (c_{1} \sin \left (\frac {\sqrt {7}\, \left (x -1\right ) \left (x +1\right ) \left (\ln \left (x^{2}-1\right )-1\right )}{2}\right )+c_{2} \cos \left (\frac {\sqrt {7}\, \left (x -1\right ) \left (x +1\right ) \left (\ln \left (x^{2}-1\right )-1\right )}{2}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\sqrt {7}\, \left (c_{3} \cos \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )-\sin \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )\right ) x}{7 c_{3} \sin \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )+7 \cos \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\sqrt {7}\, \left (c_{3} \cos \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )-\sin \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )\right ) x}{7 c_{3} \sin \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )+7 \cos \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\sqrt {7}\, \left (c_{3} \cos \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )-\sin \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )\right ) x}{7 c_{3} \sin \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )+7 \cos \left (\frac {\sqrt {7}\, \left (\ln \left (x^{2}-1\right )-1\right ) \left (x^{2}-1\right )}{2}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x y^{2}+\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x^{3}-y^{\prime } x +y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y-\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x^{3}-7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x y^{2}}{x} \end {array} \]
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 homogeneous D <- homogeneous successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 44
dsolve(diff(y(x),x) = (y(x)+ln((x+1)*(x-1))*x^3+7*ln((x+1)*(x-1))*x*y(x)^2)/x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\tan \left (\frac {\left (x^{2} \ln \left (x^{2}-1\right )-x^{2}-\ln \left (x^{2}-1\right )+2 c_{1} +1\right ) \sqrt {7}}{2}\right ) x \sqrt {7}}{7} \]
✓ Solution by Mathematica
Time used: 1.585 (sec). Leaf size: 62
DSolve[y'[x] == (x^3*Log[(-1 + x)*(1 + x)] + y[x] + 7*x*Log[(-1 + x)*(1 + x)]*y[x]^2)/x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x \tan \left (\frac {1}{2} \sqrt {7} \left (-x^2+x^2 \log (x-1)+x^2 \log (x+1)-\log (1-x)-\log (x+1)+2 c_1\right )\right )}{\sqrt {7}} \]