77.1.102 problem 129 (page 179)
Internal
problem
ID
[17992]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
129
(page
179)
Date
solved
:
Tuesday, January 28, 2025 at 11:19:33 AM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} {y^{\prime \prime }}^{2}-2 x y^{\prime \prime }-y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.215 (sec). Leaf size: 205
dsolve(diff(y(x),x$2)^2-2*x*diff(y(x),x$2)-diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \int \left (-x^{2}+\operatorname {RootOf}\left (8 \textit {\_Z}^{5}+20 x \,\textit {\_Z}^{4}-10 x^{2} \textit {\_Z}^{3}-45 x^{3} \textit {\_Z}^{2}+27 x^{5}-c_{1} \right )^{2}\right )d x +c_{2} \\
y &= \int \left (-x^{2}+\operatorname {RootOf}\left (8 \textit {\_Z}^{5}+20 x \,\textit {\_Z}^{4}-10 x^{2} \textit {\_Z}^{3}-45 x^{3} \textit {\_Z}^{2}+27 x^{5}+c_{1} \right )^{2}\right )d x +c_{2} \\
y &= \int \left (-x^{2}+\operatorname {RootOf}\left (8 \textit {\_Z}^{5}-20 x \,\textit {\_Z}^{4}-10 x^{2} \textit {\_Z}^{3}+45 x^{3} \textit {\_Z}^{2}-27 x^{5}-c_{1} \right )^{2}\right )d x +c_{2} \\
y &= \int \left (-x^{2}+\operatorname {RootOf}\left (8 \textit {\_Z}^{5}-20 x \,\textit {\_Z}^{4}-10 x^{2} \textit {\_Z}^{3}+45 x^{3} \textit {\_Z}^{2}-27 x^{5}+c_{1} \right )^{2}\right )d x +c_{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.191 (sec). Leaf size: 501
DSolve[D[y[x],{x,2}]^2-2*x*D[y[x],{x,2}]-D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[1]^5+300 \text {$\#$1}^3 K[1]^4-10 e^{5 c_1} \text {$\#$1} K[1]^3-240 \text {$\#$1}^4 K[1]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[1]-125 K[1]^6\right ) \text {$\#$1}^2\&,1\right ]dK[1]+c_2 \\
y(x)\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[2]^5+300 \text {$\#$1}^3 K[2]^4-10 e^{5 c_1} \text {$\#$1} K[2]^3-240 \text {$\#$1}^4 K[2]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[2]-125 K[2]^6\right ) \text {$\#$1}^2\&,2\right ]dK[2]+c_2 \\
y(x)\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[3]^5+300 \text {$\#$1}^3 K[3]^4-10 e^{5 c_1} \text {$\#$1} K[3]^3-240 \text {$\#$1}^4 K[3]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[3]-125 K[3]^6\right ) \text {$\#$1}^2\&,3\right ]dK[3]+c_2 \\
y(x)\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[4]^5+300 \text {$\#$1}^3 K[4]^4-10 e^{5 c_1} \text {$\#$1} K[4]^3-240 \text {$\#$1}^4 K[4]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[4]-125 K[4]^6\right ) \text {$\#$1}^2\&,4\right ]dK[4]+c_2 \\
y(x)\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[5]^5+300 \text {$\#$1}^3 K[5]^4-10 e^{5 c_1} \text {$\#$1} K[5]^3-240 \text {$\#$1}^4 K[5]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[5]-125 K[5]^6\right ) \text {$\#$1}^2\&,5\right ]dK[5]+c_2 \\
\end{align*}