77.1.100 problem 128 (page 179)

Internal problem ID [17990]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 128 (page 179)
Date solved : Tuesday, January 28, 2025 at 08:28:25 PM
CAS classification : [[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]]

\begin{align*} 40 {y^{\prime \prime \prime }}^{3}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+9 {y^{\prime \prime }}^{2} y^{\left (5\right )}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.396 (sec). Leaf size: 113

dsolve(40*diff(y(x),x$3)^3-45*diff(y(x),x$2)*diff(y(x),x$3)*diff(y(x),x$4)+9*diff(y(x),x$2)^2*diff(y(x),x$5)=0,y(x), singsol=all)
 

\begin{align*} y &= c_{1} x +c_{2} \\ y &= \int \left (\int \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{\operatorname {RootOf}\left (-\ln \left (\textit {\_f} \right )-6 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_k}}{\sqrt {\textit {\_k}^{4}-c_{1}}}d \textit {\_k} \right )+c_{2} \right ) \textit {\_f}^{{3}/{2}}}d \textit {\_f} +x +c_{3} \right )d x \right )d x +c_4 x +c_5 \\ y &= \int \left (\int \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{\operatorname {RootOf}\left (-\ln \left (\textit {\_f} \right )+6 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_k}}{\sqrt {\textit {\_k}^{4}-c_{1}}}d \textit {\_k} \right )+c_{2} \right ) \textit {\_f}^{{3}/{2}}}d \textit {\_f} +x +c_{3} \right )d x \right )d x +c_4 x +c_5 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.154 (sec). Leaf size: 43

DSolve[40*D[y[x],{x,3}]^3-45*D[y[x],{x,2}]*D[y[x],{x,3}]*D[y[x],{x,4}]+9*D[y[x],{x,2}]^2*D[y[x],{x,5}]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_5 x-\frac {4 \sqrt {x (c_3 x+c_2)+c_1}}{c_2{}^2-4 c_1 c_3}+c_4 \]