54.2.6 problem 6
Internal
problem
ID
[8538]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
16.
Nonlinear
equations.
Miscellaneous
Exercises.
Page
340
Problem
number
:
6
Date
solved
:
Monday, January 27, 2025 at 04:11:22 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} 5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.030 (sec). Leaf size: 85
dsolve(5*diff(y(x),x)^2+6*x*diff(y(x),x)-2*y(x)=0,y(x), singsol=all)
\begin{align*}
\frac {c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y}\right )^{{3}/{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+10 y}}{5} &= 0 \\
\frac {c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y}\right )^{{3}/{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+10 y}}{5} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 14.279 (sec). Leaf size: 771
DSolve[5*(D[y[x],x])^2+6*x*D[y[x],x]-2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {Root}\left [4 \text {$\#$1}^5+4 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-25000 e^{10 c_1}\&,1\right ] \\
y(x)\to \text {Root}\left [4 \text {$\#$1}^5+4 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-25000 e^{10 c_1}\&,2\right ] \\
y(x)\to \text {Root}\left [4 \text {$\#$1}^5+4 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-25000 e^{10 c_1}\&,3\right ] \\
y(x)\to \text {Root}\left [4 \text {$\#$1}^5+4 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-25000 e^{10 c_1}\&,4\right ] \\
y(x)\to \text {Root}\left [4 \text {$\#$1}^5+4 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-25000 e^{10 c_1}\&,5\right ] \\
y(x)\to \text {Root}\left [100000 \text {$\#$1}^5+100000 \text {$\#$1}^4 x^2+25000 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,1\right ] \\
y(x)\to \text {Root}\left [100000 \text {$\#$1}^5+100000 \text {$\#$1}^4 x^2+25000 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,2\right ] \\
y(x)\to \text {Root}\left [100000 \text {$\#$1}^5+100000 \text {$\#$1}^4 x^2+25000 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,3\right ] \\
y(x)\to \text {Root}\left [100000 \text {$\#$1}^5+100000 \text {$\#$1}^4 x^2+25000 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,4\right ] \\
y(x)\to \text {Root}\left [100000 \text {$\#$1}^5+100000 \text {$\#$1}^4 x^2+25000 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,5\right ] \\
y(x)\to 0 \\
\end{align*}