Internal problem ID [8649]
Internal file name [OUTPUT/7582_Sunday_June_05_2022_11_08_11_PM_5294071/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 313.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational]
Unable to solve or complete the solution.
\[ \boxed {\left (2 a y^{3}+3 a y^{2} x -b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 y b \,x^{2}=-2 b \,x^{3}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (2 a y^{3}+3 a y^{2} x -b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 y b \,x^{2}=-2 b \,x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a y^{3}-c y^{2}-3 y b \,x^{2}-2 b \,x^{3}}{2 a y^{3}+3 a y^{2} x -b \,x^{3}+c \,x^{2}} \end {array} \]
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 Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 2`[0, (x^2+2*x*y+y^2)/(3*a*x*y^2+2*a*y^3-b*x^3+c*x^2)], [0, (a*x*y^3+a*y^4+b*x^4+
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 595
dsolve((2*a*y(x)^3+3*a*x*y(x)^2-b*x^3+c*x^2)*diff(y(x),x)-a*y(x)^3+c*y(x)^2+3*b*x^2*y(x)+2*b*x^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\left (-{\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {2}{3}}+\left (c x -c_{1} \right ) a 12^{\frac {1}{3}}\right ) 12^{\frac {1}{3}}}{6 {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {1}{3}} a} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {2}{3}}+a \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (c x -c_{1} \right ) 2^{\frac {2}{3}}\right )}{12 {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {1}{3}} a} \\ y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) a \left (c x -c_{1} \right ) 2^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{12 {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {1}{3}} a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.244 (sec). Leaf size: 520
DSolve[2*b*x^3 + 3*b*x^2*y[x] + c*y[x]^2 - a*y[x]^3 + (c*x^2 - b*x^3 + 3*a*x*y[x]^2 + 2*a*y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-\sqrt [3]{2} \left (\sqrt {3} \sqrt {a^3 \left (27 a x^2 \left (b x^2+c_1\right ){}^2+4 (c x+c_1){}^3\right )}+9 a^2 b x^3+9 a^2 c_1 x\right ){}^{2/3}+2 \sqrt [3]{3} a c x+2 \sqrt [3]{3} a c_1}{6^{2/3} a \sqrt [3]{\sqrt {3} \sqrt {a^3 \left (27 a x^2 \left (b x^2+c_1\right ){}^2+4 (c x+c_1){}^3\right )}+9 a^2 b x^3+9 a^2 c_1 x}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}{6 \sqrt [3]{2} a}-\frac {i \left (\sqrt {3}-i\right ) (c x+c_1)}{2^{2/3} \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) (c x+c_1)}{2^{2/3} \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}{6 \sqrt [3]{2} a} \\ \end{align*}