Internal problem ID [9159]
Internal file name [OUTPUT/8094_Monday_June_06_2022_01_44_23_AM_23924214/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 825.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[_Abel]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (\left (x^{2}+1\right )^{\frac {3}{2}} x^{2}+\left (x^{2}+1\right )^{\frac {3}{2}}+y^{2} \left (x^{2}+1\right )^{\frac {3}{2}}+y^{3} x^{2}+y^{3}\right ) x}{\left (x^{2}+1\right )^{3}}=0} \]
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 (x^{2}+1\right ) y^{3}}{x^{6}+3 x^{4}+3 x^{2}+1}+\frac {x \left (x^{2}+1\right )^{\frac {3}{2}} y^{2}}{x^{6}+3 x^{4}+3 x^{2}+1}+\frac {\left (\left (x^{2}+1\right )^{\frac {3}{2}} x^{2}+\left (x^{2}+1\right )^{\frac {3}{2}}\right ) x}{x^{6}+3 x^{4}+3 x^{2}+1}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= \frac {x^{3} \left (x^{2}+1\right )^{\frac {3}{2}}}{x^{6}+3 x^{4}+3 x^{2}+1}+\frac {x \left (x^{2}+1\right )^{\frac {3}{2}}}{x^{6}+3 x^{4}+3 x^{2}+1}\\ f_1(x) &= 0\\ f_2(x) &= \frac {x \left (x^{2}+1\right )^{\frac {3}{2}}}{x^{6}+3 x^{4}+3 x^{2}+1}\\ f_3(x) &= \frac {x^{3}}{x^{6}+3 x^{4}+3 x^{2}+1}+\frac {x}{x^{6}+3 x^{4}+3 x^{2}+1} \end {align*}
Since \(f_2(x)=\frac {x \left (x^{2}+1\right )^{\frac {3}{2}}}{x^{6}+3 x^{4}+3 x^{2}+1}\) 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 {x \left (x^{2}+1\right )^{\frac {3}{2}}}{x^{6}+3 x^{4}+3 x^{2}+1}}{\frac {3 x^{3}}{x^{6}+3 x^{4}+3 x^{2}+1}+\frac {3 x}{x^{6}+3 x^{4}+3 x^{2}+1}} \right ) \\ &= u \left (x \right )-\frac {\sqrt {x^{2}+1}}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \frac {x^{7} u \left (x \right )^{3}}{\left (x^{2}+1\right )^{2} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}-\frac {x^{9} u \left (x \right )}{3 \left (x^{2}+1\right )^{2} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {2 x \left (x^{2}+1\right )^{\frac {5}{2}}}{27 \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {x^{7}}{3 \sqrt {x^{2}+1}\, \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {x^{9}}{\left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {3 x^{5} u \left (x \right )^{3}}{\left (x^{2}+1\right )^{2} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}-\frac {4 x^{7} u \left (x \right )}{3 \left (x^{2}+1\right )^{2} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {x^{5}}{\sqrt {x^{2}+1}\, \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {4 x^{7}}{\left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {3 x^{3} u \left (x \right )^{3}}{\left (x^{2}+1\right )^{2} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}-\frac {2 x^{5} u \left (x \right )}{\left (x^{2}+1\right )^{2} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {x^{3}}{\sqrt {x^{2}+1}\, \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {6 x^{5}}{\left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {x u \left (x \right )^{3}}{\left (x^{2}+1\right )^{2} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}-\frac {4 x^{3} u \left (x \right )}{3 \left (x^{2}+1\right )^{2} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {x}{3 \sqrt {x^{2}+1}\, \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {4 x^{3}}{\left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}-\frac {x u \left (x \right )}{3 \left (x^{2}+1\right )^{2} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}+\frac {x}{\left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}\tag {2} \end {align*}
This is Abel first kind ODE, it has the form \[ u^{\prime }\left (x \right )= f_0(x)+f_1(x) u \left (x \right ) +f_2(x)u \left (x \right )^{2}+f_3(x)u \left (x \right )^{3} \] Comparing the above to given ODE which is \begin {align*} u^{\prime }\left (x \right )&=\frac {x u \left (x \right )^{3}}{x^{4}+2 x^{2}+1}+\frac {x \left (-9 \left (x^{2}+1\right )^{\frac {3}{2}} x^{2}-9 \left (x^{2}+1\right )^{\frac {3}{2}}\right ) u \left (x \right )}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {x \left (38 x^{6}+114 x^{4}+114 x^{2}+38\right )}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= \frac {38 x^{7}}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x^{5}}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x^{3}}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}\\ f_1(x) &= -\frac {x^{3}}{3 \left (x^{4}+2 x^{2}+1\right )}-\frac {x}{3 \left (x^{4}+2 x^{2}+1\right )}\\ f_2(x) &= 0\\ f_3(x) &= \frac {x}{x^{4}+2 x^{2}+1} \end {align*}
Since \(f_2(x)=0\) then we check the Abel invariant to see if it depends on \(x\) or not. The Abel invariant is given by \begin {align*} -\frac {f_{1}^{3}}{f_{0}^{2} f_{3}} \end {align*}
Which when evaluating gives \begin {align*} -\frac {{\left (-\frac {\left (\frac {266 x^{6}}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}-\frac {38 x^{8}}{9 \left (x^{2}+1\right )^{\frac {5}{2}} \left (x^{4}+2 x^{2}+1\right )}-\frac {38 x^{7} \left (4 x^{3}+4 x \right )}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )^{2}}+\frac {190 x^{4}}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}-\frac {38 x^{6}}{3 \left (x^{2}+1\right )^{\frac {5}{2}} \left (x^{4}+2 x^{2}+1\right )}-\frac {38 x^{5} \left (4 x^{3}+4 x \right )}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )^{2}}+\frac {38 x^{2}}{3 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}-\frac {38 x^{4}}{3 \left (x^{2}+1\right )^{\frac {5}{2}} \left (x^{4}+2 x^{2}+1\right )}-\frac {38 x^{3} \left (4 x^{3}+4 x \right )}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )^{2}}+\frac {38}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}-\frac {38 x^{2}}{9 \left (x^{2}+1\right )^{\frac {5}{2}} \left (x^{4}+2 x^{2}+1\right )}-\frac {38 x \left (4 x^{3}+4 x \right )}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )^{2}}\right ) x}{x^{4}+2 x^{2}+1}+\left (\frac {38 x^{7}}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x^{5}}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x^{3}}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}\right ) \left (\frac {1}{x^{4}+2 x^{2}+1}-\frac {x \left (4 x^{3}+4 x \right )}{\left (x^{4}+2 x^{2}+1\right )^{2}}\right )+\frac {3 \left (\frac {38 x^{7}}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x^{5}}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x^{3}}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}\right ) x \left (-\frac {x^{3}}{3 \left (x^{4}+2 x^{2}+1\right )}-\frac {x}{3 \left (x^{4}+2 x^{2}+1\right )}\right )}{x^{4}+2 x^{2}+1}\right )}^{3} \left (x^{4}+2 x^{2}+1\right )^{4}}{27 x^{4} \left (\frac {38 x^{7}}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x^{5}}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x^{3}}{9 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}+\frac {38 x}{27 \left (x^{2}+1\right )^{\frac {3}{2}} \left (x^{4}+2 x^{2}+1\right )}\right )^{5}} \end {align*}
Since the Abel invariant depends on \(x\) then unable to solve this ode at this time.
Unable to complete the solution now.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{\prime } x^{6}+\left (x^{2}+1\right )^{\frac {3}{2}} y^{2} x +\left (x^{2}+1\right )^{\frac {3}{2}} x^{3}+y^{3} x^{3}-3 y^{\prime } x^{4}+\left (x^{2}+1\right )^{\frac {3}{2}} x +y^{3} x -3 x^{2} y^{\prime }-y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\left (x^{2}+1\right )^{\frac {3}{2}} y^{2} x +\left (x^{2}+1\right )^{\frac {3}{2}} x^{3}+y^{3} x^{3}+\left (x^{2}+1\right )^{\frac {3}{2}} x +y^{3} x}{-x^{6}-3 x^{4}-3 x^{2}-1} \end {array} \]
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: 48
dsolve(diff(y(x),x) = ((x^2+1)^(3/2)*x^2+(x^2+1)^(3/2)+y(x)^2*(x^2+1)^(3/2)+x^2*y(x)^3+y(x)^3)*x/(x^2+1)^3,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {x^{2}+1}\, \left (19 \operatorname {RootOf}\left (-1296 \left (\int _{}^{\textit {\_Z}}\frac {1}{361 \textit {\_a}^{3}-432 \textit {\_a} +432}d \textit {\_a} \right )+2 \ln \left (x^{2}+1\right )+3 c_{1} \right )-6\right )}{18} \]
✓ Solution by Mathematica
Time used: 1.381 (sec). Leaf size: 148
DSolve[y'[x] == (x*((1 + x^2)^(3/2) + x^2*(1 + x^2)^(3/2) + (1 + x^2)^(3/2)*y[x]^2 + y[x]^3 + x^2*y[x]^3))/(1 + x^2)^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {19}{3} \text {RootSum}\left [-19 \text {$\#$1}^3+6 \sqrt [3]{38} \text {$\#$1}-19\&,\frac {\log \left (\frac {\frac {3 x y(x)}{\left (x^2+1\right )^2}+\frac {x}{\left (x^2+1\right )^{3/2}}}{\sqrt [3]{38} \sqrt [3]{\frac {x^3}{\left (x^2+1\right )^{9/2}}}}-\text {$\#$1}\right )}{2 \sqrt [3]{38}-19 \text {$\#$1}^2}\&\right ]=\frac {19^{2/3} \left (\frac {x^3}{\left (x^2+1\right )^{9/2}}\right )^{2/3} \left (x^2+1\right )^3 \log \left (x^2+1\right )}{9 \sqrt [3]{2} x^2}+c_1,y(x)\right ] \]