Internal problem ID [9173]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1594.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-6 y^{2}+4 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 59
dsolve(diff(diff(y(x),x),x)-6*y(x)^2+4*y(x)=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {1}{\sqrt {4 \textit {\_a}^{3}-4 \textit {\_a}^{2}+c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {1}{\sqrt {4 \textit {\_a}^{3}-4 \textit {\_a}^{2}+c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.289 (sec). Leaf size: 373
DSolve[4*y[x] - 6*y[x]^2 + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {4 \left (\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,3\right ]\right ) \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,1\right ]\right ) \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,2\right ]\right ) \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,3\right ]\right ) F\left (\text {ArcSin}\left (\sqrt {\frac {\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,3\right ]-y(x)}{\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,2\right ]}}\right )|\frac {\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,1\right ]-\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,3\right ]}\right ){}^2}{\left (4 y(x)^3-4 y(x)^2+c_1\right ) \left (\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,1\right ]\right ) \left (\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-4 \text {$\#$1}^2+c_1\&,2\right ]\right )}=(x+c_2){}^2,y(x)\right ] \]