Internal problem ID [9050]
Internal file name [OUTPUT/7985_Monday_June_06_2022_01_07_57_AM_77170307/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 716.
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.
\[ \boxed {y^{\prime }-\frac {3 x^{4}+3 x^{3}+\sqrt {9 x^{4}-4 y^{3}}}{\left (1+x \right ) y^{2}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{\prime } y^{2} x +3 x^{4}-y^{2} y^{\prime }+3 x^{3}+\sqrt {9 x^{4}-4 y^{3}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-3 x^{4}-3 x^{3}-\sqrt {9 x^{4}-4 y^{3}}}{-y^{2} x -y^{2}} \end {array} \]
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 = 2 `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 5`[0, (9*x^4-4*y^3)^(1/2)/y^2]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 37
dsolve(diff(y(x),x) = (3*x^4+3*x^3+(9*x^4-4*y(x)^3)^(1/2))/(x+1)/y(x)^2,y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {9 x^{4}-4 \textit {\_a}^{3}}}d \textit {\_a} -\ln \left (x +1\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 3.458 (sec). Leaf size: 133
DSolve[y'[x] == (3*x^3 + 3*x^4 + Sqrt[9*x^4 - 4*y[x]^3])/((1 + x)*y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (-\frac {3}{2}\right )^{2/3} \sqrt [3]{x^4-4 \log ^2(x+1)+8 c_1 \log (x+1)-4 c_1{}^2} \\ y(x)\to \left (\frac {3}{2}\right )^{2/3} \sqrt [3]{x^4-4 \log ^2(x+1)+8 c_1 \log (x+1)-4 c_1{}^2} \\ y(x)\to -\sqrt [3]{-1} \left (\frac {3}{2}\right )^{2/3} \sqrt [3]{x^4-4 \log ^2(x+1)+8 c_1 \log (x+1)-4 c_1{}^2} \\ \end{align*}