Internal problem ID [9238]
Internal file name [OUTPUT/8174_Monday_June_06_2022_02_02_31_AM_5196146/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 905.
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]
\[ \boxed {y^{\prime }-\frac {a^{2} x +a^{3} x^{3}+y^{2} a^{3} x^{3}+2 y a^{2} x^{2}+a x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 a x y+1}{a^{3} x^{3}}=0} \]
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 }&=y^{3}+\frac {\left (a^{3} x^{3}+3 a^{2} x^{2}\right ) y^{2}}{a^{3} x^{3}}+\frac {\left (2 a^{2} x^{2}+3 a x \right ) y}{a^{3} x^{3}}+\frac {a^{3} x^{3}+a^{2} x +a x +1}{a^{3} x^{3}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= 1+\frac {1}{a \,x^{2}}+\frac {1}{a^{2} x^{2}}+\frac {1}{a^{3} x^{3}}\\ f_1(x) &= \frac {2}{a x}+\frac {3}{a^{2} x^{2}}\\ f_2(x) &= 1+\frac {3}{a x}\\ f_3(x) &= 1 \end {align*}
Since \(f_2(x)=1+\frac {3}{a x}\) 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 {1+\frac {3}{a x}}{3} \right ) \\ &= u \left (x \right )-\frac {1}{3}-\frac {1}{a x} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = u \left (x \right )^{3}-\frac {u \left (x \right )}{3}+\frac {29}{27}\tag {2} \end {align*}
The above ODE (2) can now be solved as separable.
Integrating both sides gives \begin {align*} \int _{}^{u \left (x \right )}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \end {align*}
Substituting \(u=y-\frac {1}{3}-\frac {1}{a x}\) in the above solution gives \begin {align*} \int _{}^{y-\frac {1}{3}-\frac {1}{a x}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y-\frac {1}{3}-\frac {1}{a x}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y-\frac {1}{3}-\frac {1}{a x}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{3} a^{3} x^{3}-y^{2} a^{3} x^{3}+y^{\prime } a^{3} x^{3}-3 y^{2} a^{2} x^{2}-a^{3} x^{3}-2 y a^{2} x^{2}-3 a x y-a^{2} x -a x -1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a^{2} x +a^{3} x^{3}+y^{2} a^{3} x^{3}+2 y a^{2} x^{2}+a x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 a x y+1}{a^{3} x^{3}} \end {array} \]
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: 46
dsolve(diff(y(x),x) = (a^2*x+a^3*x^3+a^3*x^3*y(x)^2+2*a^2*x^2*y(x)+a*x+y(x)^3*a^3*x^3+3*y(x)^2*a^2*x^2+3*y(x)*a*x+1)/a^3/x^3,y(x), singsol=all)
\[ y \left (x \right ) = \frac {29 \operatorname {RootOf}\left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+x +3 c_{1} \right ) a x -3 a x -9}{9 a x} \]
✓ Solution by Mathematica
Time used: 0.185 (sec). Leaf size: 85
DSolve[y'[x] == (1 + a*x + a^2*x + a^3*x^3 + 3*a*x*y[x] + 2*a^2*x^2*y[x] + 3*a^2*x^2*y[x]^2 + a^3*x^3*y[x]^2 + a^3*x^3*y[x]^3)/(a^3*x^3),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {\frac {a x+3}{a x}+3 y(x)}{\sqrt [3]{29}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=\frac {1}{9} 29^{2/3} x+c_1,y(x)\right ] \]