Internal problem ID [9378]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1804 (book 6.213).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {2 \left (c -y\right ) \left (b -y\right ) \left (a -y\right ) y^{\prime \prime }+\left (\left (a -y\right ) \left (b -y\right )+\left (c -y\right ) \left (a -y\right )+\left (b -y\right ) \left (c -y\right )\right ) {y^{\prime }}^{2}-\operatorname {a0} \left (c -y\right )^{2} \left (b -y\right )^{2} \left (a -y\right )^{2}-\operatorname {a1} \left (b -y\right )^{2} \left (c -y\right )^{2}-\operatorname {a2} \left (c -y\right )^{2} \left (a -y\right )^{2}-\operatorname {a3} \left (b -y\right )^{2} \left (a -y\right )^{2}=0} \]
✓ Solution by Maple
Time used: 38.922 (sec). Leaf size: 304
dsolve(2*(c-y(x))*(b-y(x))*(a-y(x))*diff(diff(y(x),x),x)+((a-y(x))*(b-y(x))+(c-y(x))*(a-y(x))+(b-y(x))*(c-y(x)))*diff(y(x),x)^2-a0*(c-y(x))^2*(b-y(x))^2*(a-y(x))^2-a1*(b-y(x))^2*(c-y(x))^2-a2*(c-y(x))^2*(a-y(x))^2-a3*(b-y(x))^2*(a-y(x))^2=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\operatorname {a0} \,\textit {\_a}^{4}+\textit {\_a}^{3} a \operatorname {a0} +\textit {\_a}^{3} \operatorname {a0} b +\textit {\_a}^{3} \operatorname {a0} c -\textit {\_a}^{2} a \operatorname {a0} b -\textit {\_a}^{2} a \operatorname {a0} c -\textit {\_a}^{2} \operatorname {a0} b c +\textit {\_a} a \operatorname {a0} b c +c_{1} \textit {\_a}^{3}-c_{1} \textit {\_a}^{2} a -c_{1} \textit {\_a}^{2} b -c_{1} \textit {\_a}^{2} c +c_{1} \textit {\_a} a b +c_{1} \textit {\_a} a c +c_{1} \textit {\_a} b c -c_{1} a b c +\textit {\_a}^{2} \operatorname {a1} +\textit {\_a}^{2} \operatorname {a2} +\textit {\_a}^{2} \operatorname {a3} -\textit {\_a} a \operatorname {a2} -\textit {\_a} a \operatorname {a3} -\textit {\_a} \operatorname {a1} b -\textit {\_a} \operatorname {a1} c -\textit {\_a} \operatorname {a2} c -\textit {\_a} \operatorname {a3} b +a \operatorname {a2} c +a \operatorname {a3} b +b c \operatorname {a1}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {\left (-\left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right ) c_{1} +\textit {\_a} \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right ) \operatorname {a0} +\operatorname {a1} b +\left (-\operatorname {a1} -\operatorname {a2} \right ) \textit {\_a} +a \operatorname {a2} \right ) c +\textit {\_a} \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right ) c_{1} -\textit {\_a}^{2} \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right ) \operatorname {a0} +\left (\left (-\operatorname {a1} -\operatorname {a3} \right ) \textit {\_a} +a \operatorname {a3} \right ) b -\textit {\_a} \left (\left (-\operatorname {a1} -\operatorname {a2} -\operatorname {a3} \right ) \textit {\_a} +a \left (\operatorname {a2} +\operatorname {a3} \right )\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 17.39 (sec). Leaf size: 10387
DSolve[-(a3*(a - y[x])^2*(b - y[x])^2) - a2*(a - y[x])^2*(c - y[x])^2 - a1*(b - y[x])^2*(c - y[x])^2 - a0*(a - y[x])^2*(b - y[x])^2*(c - y[x])^2 + ((a - y[x])*(b - y[x]) + (a - y[x])*(c - y[x]) + (b - y[x])*(c - y[x]))*y'[x]^2 + 2*(a - y[x])*(b - y[x])*(c - y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
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