Internal problem ID [9300]
Internal file name [OUTPUT/8236_Monday_June_06_2022_02_26_04_AM_10804671/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 967.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]
\[ \boxed {y^{\prime }+\frac {x \left (-513-432 x -1296 y^{2} x^{2}+432 y^{2} x^{3}-864 x^{4}-378 y-540 y^{2}-216 y^{3} x^{6}-456 x^{6}-216 y^{3}-756 x^{3}-216 x^{4} y-1134 x^{2}-144 x^{7}+720 y x^{3}-594 y x^{2}+64 x^{9}-96 x^{8}-288 x^{6} y-576 x^{5}-972 y^{2} x^{4}+1008 y x^{5}-288 y x^{8}-648 y^{3} x^{2}+288 y x^{7}-648 y^{3} x^{4}+864 y^{2} x^{5}-216 y^{2} x^{6}+432 y^{2} x^{7}\right )}{216 \left (x^{2}+1\right )^{4}}=0} \]
This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(x)+f_1(x) y +f_2(x)y^{2}+f_3(x)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=\frac {x \left (216 x^{6}+648 x^{4}+648 x^{2}+216\right ) y^{3}}{216 x^{8}+864 x^{6}+1296 x^{4}+864 x^{2}+216}+\frac {x \left (-432 x^{7}+216 x^{6}-864 x^{5}+972 x^{4}-432 x^{3}+1296 x^{2}+540\right ) y^{2}}{216 x^{8}+864 x^{6}+1296 x^{4}+864 x^{2}+216}+\frac {x \left (288 x^{8}-288 x^{7}+288 x^{6}-1008 x^{5}+216 x^{4}-720 x^{3}+594 x^{2}+378\right ) y}{216 x^{8}+864 x^{6}+1296 x^{4}+864 x^{2}+216}+\frac {x \left (-64 x^{9}+96 x^{8}+144 x^{7}+456 x^{6}+576 x^{5}+864 x^{4}+756 x^{3}+1134 x^{2}+432 x +513\right )}{216 x^{8}+864 x^{6}+1296 x^{4}+864 x^{2}+216}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= -\frac {8 x^{10}}{27 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {4 x^{9}}{9 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {2 x^{8}}{3 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {19 x^{7}}{9 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {8 x^{6}}{3 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {4 x^{5}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {7 x^{4}}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {21 x^{3}}{4 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {2 x^{2}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {19 x}{8 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}\\ f_1(x) &= \frac {7 x}{4 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {4 x^{9}}{3 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {4 x^{8}}{3 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {4 x^{7}}{3 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {14 x^{6}}{3 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {x^{5}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}-\frac {10 x^{4}}{3 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}+\frac {11 x^{3}}{4 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}\\ f_2(x) &= \frac {5 x}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{8}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}-\frac {4 x^{6}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{4}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {6 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}\\ f_3(x) &= \frac {x}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {3 x^{5}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {3 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1} \end {align*}
Since \(f_2(x)=\frac {5 x}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{8}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}-\frac {4 x^{6}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{4}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {6 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}\) is not zero, then the first step is to apply the following transformation to remove \(f_2\). Let \(y = u(x) - \frac {f_2}{3 f_3}\) or \begin {align*} y &= u(x) - \left ( \frac {\frac {5 x}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{8}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}-\frac {4 x^{6}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{4}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {6 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}}{\frac {3 x}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {3 x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}} \right ) \\ &= u \left (x \right )+\frac {4 x^{3}-2 x^{2}-5}{6 x^{2}+6} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \frac {x u \left (x \right )^{3}}{x^{2}+1}-\frac {x u \left (x \right )}{3 \left (x^{2}+1\right )}+\frac {29 x}{27 \left (x^{2}+1\right )}\tag {2} \end {align*}
The above ODE (2) can now be solved as separable.
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= \frac {x \left (u^{3}-\frac {1}{3} u +\frac {29}{27}\right )}{x^{2}+1} \end {align*}
Where \(f(x)=\frac {x}{x^{2}+1}\) and \(g(u)=u^{3}-\frac {1}{3} u +\frac {29}{27}\). Integrating both sides gives \begin{align*} \frac {1}{u^{3}-\frac {1}{3} u +\frac {29}{27}} \,du &= \frac {x}{x^{2}+1} \,d x \\ \int { \frac {1}{u^{3}-\frac {1}{3} u +\frac {29}{27}} \,du} &= \int {\frac {x}{x^{2}+1} \,d x} \\ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}&=\frac {\ln \left (x^{2}+1\right )}{2}+c_{2} \\ \end{align*} Which results in \[ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}=\frac {\ln \left (x^{2}+1\right )}{2}+c_{2} \] The solution is \[ \int _{}^{u \left (x \right )}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {\ln \left (x^{2}+1\right )}{2}-c_{2} = 0 \] Substituting \(u=y+\frac {\frac {5 x}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{8}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}-\frac {4 x^{6}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{4}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {6 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}}{\frac {3 x}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {3 x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}}\) in the above solution gives \begin {align*} \int _{}^{y+\frac {\frac {5 x}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{8}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}-\frac {4 x^{6}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{4}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {6 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}}{\frac {3 x}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {3 x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {\ln \left (x^{2}+1\right )}{2}-c_{2} = 0 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y+\frac {\frac {5 x}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{8}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}-\frac {4 x^{6}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{4}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {6 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}}{\frac {3 x}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {3 x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {\ln \left (x^{2}+1\right )}{2}-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ \int _{}^{y+\frac {\frac {5 x}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{8}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}-\frac {4 x^{6}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{4}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {6 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}}{\frac {3 x}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {3 x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {\ln \left (x^{2}+1\right )}{2}-c_{2} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 513 x +216 x y^{3}+1296 y^{2} x^{3}+756 x^{4}-216 y^{\prime }-432 y^{2} x^{8}-216 y^{\prime } x^{8}+576 x^{6}+1134 x^{3}-720 x^{4} y+540 y^{2} x +432 x^{2}+456 x^{7}+594 y x^{3}+96 x^{9}+144 x^{8}-1008 x^{6} y-864 x^{2} y^{\prime }+378 y x +216 y^{3} x^{7}-1296 y^{\prime } x^{4}+864 x^{5}+648 y^{3} x^{3}-432 y^{2} x^{4}+216 y x^{5}-288 y x^{8}-64 x^{10}+288 y x^{7}+972 y^{2} x^{5}-864 y^{2} x^{6}+288 y x^{9}-864 y^{\prime } x^{6}+648 x^{5} y^{3}+216 y^{2} x^{7}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-513 x -216 x y^{3}-1296 y^{2} x^{3}-756 x^{4}+432 y^{2} x^{8}-576 x^{6}-1134 x^{3}+720 x^{4} y-540 y^{2} x -432 x^{2}-456 x^{7}-594 y x^{3}-96 x^{9}-144 x^{8}+1008 x^{6} y-378 y x -216 y^{3} x^{7}-864 x^{5}-648 y^{3} x^{3}+432 y^{2} x^{4}-216 y x^{5}+288 y x^{8}+64 x^{10}-288 y x^{7}-972 y^{2} x^{5}+864 y^{2} x^{6}-288 y x^{9}-648 x^{5} y^{3}-216 y^{2} x^{7}}{-216 x^{8}-864 x^{6}-1296 x^{4}-864 x^{2}-216} \end {array} \]
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 trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 91
dsolve(diff(y(x),x) = -1/216*x/(x^2+1)^4*(-513-432*x-288*y(x)*x^8+288*y(x)*x^7-288*y(x)*x^6+864*y(x)^2*x^5-648*y(x)^3*x^4-216*y(x)*x^4-456*x^6-576*x^5+432*y(x)^2*x^7-216*y(x)^2*x^6+1008*x^5*y(x)-216*x^6*y(x)^3-972*x^4*y(x)^2+432*x^3*y(x)^2+720*x^3*y(x)-648*x^2*y(x)^3-1296*x^2*y(x)^2-594*x^2*y(x)-864*x^4-378*y(x)-216*y(x)^3-756*x^3-540*y(x)^2-1134*x^2-144*x^7+64*x^9-96*x^8),y(x), singsol=all)
\[ y \left (x \right ) = \frac {58 \operatorname {RootOf}\left (-162 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+\ln \left (x^{2}+1\right )+6 c_{1} \right ) x^{2}+12 x^{3}-6 x^{2}+58 \operatorname {RootOf}\left (-162 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+\ln \left (x^{2}+1\right )+6 c_{1} \right )-15}{18 x^{2}+18} \]
✓ Solution by Mathematica
Time used: 1.396 (sec). Leaf size: 151
DSolve[y'[x] == -1/216*(x*(-513 - 432*x - 1134*x^2 - 756*x^3 - 864*x^4 - 576*x^5 - 456*x^6 - 144*x^7 - 96*x^8 + 64*x^9 - 378*y[x] - 594*x^2*y[x] + 720*x^3*y[x] - 216*x^4*y[x] + 1008*x^5*y[x] - 288*x^6*y[x] + 288*x^7*y[x] - 288*x^8*y[x] - 540*y[x]^2 - 1296*x^2*y[x]^2 + 432*x^3*y[x]^2 - 972*x^4*y[x]^2 + 864*x^5*y[x]^2 - 216*x^6*y[x]^2 + 432*x^7*y[x]^2 - 216*y[x]^3 - 648*x^2*y[x]^3 - 648*x^4*y[x]^3 - 216*x^6*y[x]^3))/(1 + x^2)^4,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {\frac {3 x y(x)}{x^2+1}+\frac {-4 x^4+2 x^3+5 x}{2 \left (x^2+1\right )^2}}{\sqrt [3]{29} \sqrt [3]{\frac {x^3}{\left (x^2+1\right )^3}}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=\frac {29^{2/3} \left (\frac {x^3}{\left (x^2+1\right )^3}\right )^{2/3} \left (x^2+1\right )^2 \log \left (x^2+1\right )}{18 x^2}+c_1,y(x)\right ] \]