Internal problem ID [9090]
Internal file name [OUTPUT/8025_Monday_June_06_2022_01_19_22_AM_49781190/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 756.
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, _Abel]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {2 y x^{3}+x^{6}+y^{2} x^{2}+y^{3}}{x^{4}}=0} \]
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 {y^{3}}{x^{4}}+\frac {y^{2}}{x^{2}}+\frac {2 y}{x}+x^{2}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= x^{2}\\ f_1(x) &= \frac {2}{x}\\ f_2(x) &= \frac {1}{x^{2}}\\ f_3(x) &= \frac {1}{x^{4}} \end {align*}
Since \(f_2(x)=\frac {1}{x^{2}}\) 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 {1}{x^{2}}}{\frac {3}{x^{4}}} \right ) \\ &= u \left (x \right )-\frac {x^{2}}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \frac {29 x^{2}}{27}-\frac {u \left (x \right )}{3}+\frac {2 u \left (x \right )}{x}+\frac {u \left (x \right )^{3}}{x^{4}}\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 {u \left (x \right )^{3}}{x^{4}}+\frac {\left (-9 x^{4}+54 x^{3}\right ) u \left (x \right )}{27 x^{4}}+\frac {29 x^{2}}{27}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= \frac {29 x^{2}}{27}\\ f_1(x) &= -\frac {1}{3}+\frac {2}{x}\\ f_2(x) &= 0\\ f_3(x) &= \frac {1}{x^{4}} \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 {531441 \left (-\frac {58}{9 x^{3}}+\frac {-\frac {29}{27}+\frac {58}{9 x}}{x^{2}}\right )^{3} x^{6}}{20511149} \end {align*}
Since the Abel invariant depends on \(x\) then unable to solve this ode at this time.
Unable to complete the solution now.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -x^{6}+y^{\prime } x^{4}-y^{2} x^{2}-2 y x^{3}-y^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2 y x^{3}+x^{6}+y^{2} x^{2}+y^{3}}{x^{4}} \end {array} \]
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: 37
dsolve(diff(y(x),x) = (2*x^3*y(x)+x^6+x^2*y(x)^2+y(x)^3)/x^4,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-3+29 \operatorname {RootOf}\left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+x +3 c_{1} \right )\right ) x^{2}}{9} \]
✓ Solution by Mathematica
Time used: 1.14 (sec). Leaf size: 95
DSolve[y'[x] == (x^6 + 2*x^3*y[x] + x^2*y[x]^2 + y[x]^3)/x^4,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {\frac {3 y(x)}{x^4}+\frac {1}{x^2}}{\sqrt [3]{29} \sqrt [3]{\frac {1}{x^6}}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=\frac {1}{9} 29^{2/3} \left (\frac {1}{x^6}\right )^{2/3} x^5+c_1,y(x)\right ] \]