60.7.160 problem 1751 (book 6.160)
Internal
problem
ID
[11749]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1751
(book
6.160)
Date
solved
:
Monday, January 27, 2025 at 11:33:52 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} 4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+a y^{3}+b y^{2}+c y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.092 (sec). Leaf size: 90
dsolve(4*diff(diff(y(x),x),x)*y(x)-3*diff(y(x),x)^2+a*y(x)^3+y(x)^2*b+c*y(x)=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
-\sqrt {3}\, \left (\int _{}^{y}\frac {1}{\sqrt {\textit {\_a} \left (-a \,\textit {\_a}^{2}-3 b \textit {\_a} +3 c_{1} \sqrt {\textit {\_a}}+3 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\sqrt {3}\, \left (\int _{}^{y}\frac {1}{\sqrt {\textit {\_a} \left (-a \,\textit {\_a}^{2}-3 b \textit {\_a} +3 c_{1} \sqrt {\textit {\_a}}+3 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.441 (sec). Leaf size: 323
DSolve[c*y[x] + b*y[x]^2 + a*y[x]^3 - 3*D[y[x],x]^2 + 4*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {-\frac {1}{3} a K[1]^3-b K[1]^2+c_1 K[1]^{3/2}+c K[1]}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\frac {1}{3} a K[2]^3-b K[2]^2+c_1 K[2]^{3/2}+c K[2]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {-\frac {1}{3} a K[1]^3-b K[1]^2-c_1 K[1]^{3/2}+c K[1]}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {-\frac {1}{3} a K[1]^3-b K[1]^2+c_1 K[1]^{3/2}+c K[1]}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\frac {1}{3} a K[2]^3-b K[2]^2-c_1 K[2]^{3/2}+c K[2]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\frac {1}{3} a K[2]^3-b K[2]^2+c_1 K[2]^{3/2}+c K[2]}}dK[2]\&\right ][x+c_2] \\
\end{align*}