Internal problem ID [9187]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 852.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class C`], _Abel]
\[ \boxed {y^{\prime }-\frac {\alpha ^{3}+y^{2} \alpha ^{3}+2 y \alpha ^{2} \beta x +\alpha \,\beta ^{2} x^{2}+y^{3} \alpha ^{3}+3 y^{2} \alpha ^{2} \beta x +3 y \alpha \,\beta ^{2} x^{2}+\beta ^{3} x^{3}}{\alpha ^{3}}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 42
dsolve(diff(y(x),x) = (alpha^3+y(x)^2*alpha^3+2*y(x)*alpha^2*beta*x+alpha*beta^2*x^2+y(x)^3*alpha^3+3*y(x)^2*alpha^2*beta*x+3*y(x)*alpha*beta^2*x^2+beta^3*x^3)/alpha^3,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{3} \alpha +\textit {\_a}^{2} \alpha +\alpha +\beta }d \textit {\_a} \right ) \alpha -x +c_{1} \right ) \alpha -\beta x}{\alpha } \]
✓ Solution by Mathematica
Time used: 2.588 (sec). Leaf size: 902
DSolve[y'[x] == (\[Alpha]^3 + \[Alpha]*\[Beta]^2*x^2 + \[Beta]^3*x^3 + 2*\[Alpha]^2*\[Beta]*x*y[x] + 3*\[Alpha]*\[Beta]^2*x^2*y[x] + \[Alpha]^3*y[x]^2 + 3*\[Alpha]^2*\[Beta]*x*y[x]^2 + \[Alpha]^3*y[x]^3)/\[Alpha]^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{9} \text {RootSum}\left [841 \alpha ^2 \text {$\#$1}^9+729 \beta ^2 \text {$\#$1}^9+1566 \alpha \beta \text {$\#$1}^9+2523 \alpha ^2 \text {$\#$1}^6+2187 \beta ^2 \text {$\#$1}^6+4698 \alpha \beta \text {$\#$1}^6+2496 \alpha ^2 \text {$\#$1}^3+2187 \beta ^2 \text {$\#$1}^3+4698 \alpha \beta \text {$\#$1}^3+841 \alpha ^2+729 \beta ^2+1566 \alpha \beta \&,\frac {841 \alpha ^2 \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^6+729 \beta ^2 \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^6+1566 \alpha \beta \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^6+87 \alpha ^{5/3} \sqrt [3]{29 \alpha +27 \beta } \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^4+81 \alpha ^{2/3} \beta \sqrt [3]{29 \alpha +27 \beta } \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^4+1682 \alpha ^2 \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^3+1458 \beta ^2 \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^3+3132 \alpha \beta \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^3+9 \alpha ^{4/3} (29 \alpha +27 \beta )^{2/3} \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^2+87 \alpha ^{5/3} \sqrt [3]{29 \alpha +27 \beta } \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}+81 \alpha ^{2/3} \beta \sqrt [3]{29 \alpha +27 \beta } \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}+841 \alpha ^2 \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right )+729 \beta ^2 \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right )+1566 \alpha \beta \log \left (\frac {\frac {\alpha +3 x \beta }{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right )}{841 \alpha ^2 \text {$\#$1}^8+729 \beta ^2 \text {$\#$1}^8+1566 \alpha \beta \text {$\#$1}^8+1682 \alpha ^2 \text {$\#$1}^5+1458 \beta ^2 \text {$\#$1}^5+3132 \alpha \beta \text {$\#$1}^5+832 \alpha ^2 \text {$\#$1}^2+729 \beta ^2 \text {$\#$1}^2+1566 \alpha \beta \text {$\#$1}^2}\&\right ]=\frac {1}{9} \left (\frac {29 \alpha +27 \beta }{\alpha }\right )^{2/3} x+c_1,y(x)\right ] \]