Internal problem ID [9054]
Internal file name [OUTPUT/7989_Monday_June_06_2022_01_08_34_AM_4052457/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 720.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`y=_G(x,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {x^{3} \left (3 x +3+\sqrt {9 x^{4}-4 y^{3}}\right )}{\left (x +1\right ) y^{2}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y^{2} x -x^{3} \sqrt {9 x^{4}-4 y^{3}}-3 x^{4}+y^{\prime } y^{2}-3 x^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{3} \sqrt {9 x^{4}-4 y^{3}}+3 x^{4}+3 x^{3}}{y^{2} x +y^{2}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5`[0, (9*x^4-4*y^3)^(1/2)/y^2]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 48
dsolve(diff(y(x),x) = x^3*(3*x+3+(9*x^4-4*y(x)^3)^(1/2))/(x+1)/y(x)^2,y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {9 x^{4}-4 \textit {\_a}^{3}}}d \textit {\_a} -\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (x +1\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 4.423 (sec). Leaf size: 321
DSolve[y'[x] == (x^3*(3 + 3*x + Sqrt[9*x^4 - 4*y[x]^3]))/((1 + x)*y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-4 x^6+12 x^5-24 x^4+8 (-1+3 c_1) x^3-6 (-5+6 c_1) x^2+12 \left (2 x^3-3 x^2+6 x+11-6 c_1\right ) \log (x+1)-36 \log ^2(x+1)+12 (-11+6 c_1) x-(11-6 c_1){}^2}}{2^{2/3}} \\ y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{-4 x^6+12 x^5-24 x^4+8 (-1+3 c_1) x^3-6 (-5+6 c_1) x^2+12 \left (2 x^3-3 x^2+6 x+11-6 c_1\right ) \log (x+1)-36 \log ^2(x+1)+12 (-11+6 c_1) x-(11-6 c_1){}^2}}{2^{2/3}} \\ y(x)\to \left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{-4 x^6+12 x^5-24 x^4+8 (-1+3 c_1) x^3-6 (-5+6 c_1) x^2+12 \left (2 x^3-3 x^2+6 x+11-6 c_1\right ) \log (x+1)-36 \log ^2(x+1)+12 (-11+6 c_1) x-(11-6 c_1){}^2} \\ \end{align*}