Internal problem ID [4679]
Internal file name [OUTPUT/4172_Sunday_June_05_2022_12_36_12_PM_19241691/index.tex]
Book: Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section: Abel ODE’s with constant invariant
Problem number: problem 146.
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.
\[ \boxed {x^{2} y^{\prime }+x y^{3}+y^{2} a=0} \]
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}-\frac {a y^{2}}{x^{2}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= 0\\ f_1(x) &= 0\\ f_2(x) &= -\frac {a}{x^{2}}\\ f_3(x) &= -\frac {1}{x} \end {align*}
Since \(f_2(x)=-\frac {a}{x^{2}}\) 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 {a}{x^{2}}}{-\frac {3}{x}} \right ) \\ &= u \left (x \right )-\frac {a}{3 x} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = -\frac {u \left (x \right )^{3}}{x}+\frac {u \left (x \right ) a^{2}}{3 x^{3}}-\frac {2 a^{3}}{27 x^{4}}-\frac {a}{3 x^{2}}\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}+\frac {u \left (x \right ) a^{2}}{3 x^{3}}-\frac {2 a^{3}+9 a \,x^{2}}{27 x^{4}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= -\frac {2 a^{3}}{27 x^{4}}-\frac {a}{3 x^{2}}\\ f_1(x) &= \frac {a^{2}}{3 x^{3}}\\ f_2(x) &= 0\\ f_3(x) &= -\frac {1}{x} \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 {\frac {8 a^{3}}{27 x^{5}}+\frac {2 a}{3 x^{3}}}{x}+\frac {-\frac {2 a^{3}}{27 x^{4}}-\frac {a}{3 x^{2}}}{x^{2}}-\frac {\left (-\frac {2 a^{3}}{27 x^{4}}-\frac {a}{3 x^{2}}\right ) a^{2}}{x^{4}}\right )}^{3} x^{4}}{27 \left (-\frac {2 a^{3}}{27 x^{4}}-\frac {a}{3 x^{2}}\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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }+x y^{3}+y^{2} a =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x y^{3}+y^{2} a}{x^{2}} \end {array} \]
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: 65
dsolve(x^2*diff(y(x),x)+x*y(x)^3+a*y(x)^2 = 0,y(x), singsol=all)
\[ c_{1} +{\mathrm e}^{-\frac {\left (\left (a +x \right ) y \left (x \right )+x \right ) \left (\left (a -x \right ) y \left (x \right )+x \right )}{2 y \left (x \right )^{2} x^{2}}} x +\frac {\operatorname {erf}\left (\frac {\sqrt {2}\, \left (a y \left (x \right )+x \right )}{2 y \left (x \right ) x}\right ) \sqrt {2}\, \sqrt {\pi }\, a \,{\mathrm e}^{\frac {1}{2}}}{2} = 0 \]
✓ Solution by Mathematica
Time used: 0.61 (sec). Leaf size: 78
DSolve[x^2*y'[x]+x*y[x]^3+a*y[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {i a}{x}=\frac {2 e^{\frac {1}{2} \left (-\frac {i a}{x}-\frac {i}{y(x)}\right )^2}}{\sqrt {2 \pi } \text {erfi}\left (\frac {-\frac {i a}{x}-\frac {i}{y(x)}}{\sqrt {2}}\right )+2 c_1},y(x)\right ] \]