Internal problem ID [3934]
Internal file name [OUTPUT/3427_Sunday_June_05_2022_09_18_42_AM_27610018/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 687.
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.
\[ \boxed {\left (x^{2}-x^{3}+3 y^{2} x +2 y^{3}\right ) y^{\prime }+3 y x^{2}+y^{2}-y^{3}=-2 x^{3}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{2}-x^{3}+3 y^{2} x +2 y^{3}\right ) y^{\prime }+3 y x^{2}+y^{2}-y^{3}=-2 x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-2 x^{3}-3 y x^{2}-y^{2}+y^{3}}{x^{2}-x^{3}+3 y^{2} x +2 y^{3}} \end {array} \]
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 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)/(x^3-3*x*y^2-2*y^3-x^2)], [0, (x^4+x^3*y+x*y^3+y^4+x^2*y+x*
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 409
dsolve((x^2-x^3+3*x*y(x)^2+2*y(x)^3)*diff(y(x),x)+2*x^3+3*x^2*y(x)+y(x)^2-y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 x \,c_{1}^{2}+12 c_{1}^{3}}\right )^{\frac {2}{3}}-12 c_{1} -12 x}{6 \left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 x \,c_{1}^{2}+12 c_{1}^{3}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 x \,c_{1}^{2}+12 c_{1}^{3}}\right )^{\frac {2}{3}}+\left (c_{1} +x \right ) \left (i \sqrt {3}-1\right )}{\left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 x \,c_{1}^{2}+12 c_{1}^{3}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 x \,c_{1}^{2}+12 c_{1}^{3}}\right )^{\frac {2}{3}}}{12}+\left (c_{1} +x \right ) \left (1+i \sqrt {3}\right )}{\left (-108 x^{3}-108 c_{1} x +12 \sqrt {81 x^{6}+162 c_{1} x^{4}+12 x^{3}+\left (81 c_{1}^{2}+36 c_{1} \right ) x^{2}+36 x \,c_{1}^{2}+12 c_{1}^{3}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.541 (sec). Leaf size: 368
DSolve[(x^2-x^3+3 x y[x]^2+2 y[x]^3)y'[x]+2 x^3+3 x^2 y[x]+y[x]^2-y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} (x+c_1)}{\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}-\frac {\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}{3 \sqrt [3]{2}} \\ y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x\right ){}^{2/3}-6 i \sqrt [3]{2} \left (\sqrt {3}-i\right ) (x+c_1)}{12 \sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}} \\ y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x\right ){}^{2/3}+6 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) (x+c_1)}{12 \sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}} \\ y(x)\to -x \\ \end{align*}