Internal problem ID [8448]
Internal file name [OUTPUT/7381_Sunday_June_05_2022_10_54_00_PM_78474013/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 111.
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.
\[ \boxed {x y^{\prime }+y^{3}+3 x y^{2}=0} \]
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}-3 y^{2}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= 0\\ f_1(x) &= 0\\ f_2(x) &= -3\\ f_3(x) &= -\frac {1}{x} \end {align*}
Since \(f_2(x)=-3\) 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 {-3}{-\frac {3}{x}} \right ) \\ &= u \left (x \right )-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}+3 x u \left (x \right )-2 x^{2}+1\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}+3 x u \left (x \right )-\frac {2 x^{3}-x}{x}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= -2 x^{2}+1\\ f_1(x) &= 3 x\\ 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 (-13+\frac {-2 x^{2}+1}{x^{2}}+18 x^{2}\right )^{3} x^{4}}{27 \left (-2 x^{2}+1\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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }+y^{3}+3 x y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{3}-3 x y^{2}}{x} \end {array} \]
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 Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 59
dsolve(x*diff(y(x),x) + y(x)^3 + 3*x*y(x)^2=0,y(x), singsol=all)
\[ \frac {3 \,\operatorname {erf}\left (\frac {i \left (3 x y \left (x \right )-1\right ) \sqrt {2}}{2 y \left (x \right )}\right ) \sqrt {2}\, \sqrt {\pi }\, x -2 i {\mathrm e}^{\frac {\left (3 x y \left (x \right )-1\right )^{2}}{2 y \left (x \right )^{2}}}+6 c_{1} x}{6 x} = 0 \]
✓ Solution by Mathematica
Time used: 0.333 (sec). Leaf size: 55
DSolve[x*y'[x] + y[x]^3 + 3*x*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-3 x=\frac {2 e^{\frac {1}{2} \left (\frac {1}{y(x)}-3 x\right )^2}}{\sqrt {2 \pi } \text {erfi}\left (\frac {\frac {1}{y(x)}-3 x}{\sqrt {2}}\right )+2 c_1},y(x)\right ] \]