53.3.24 problem 27
Internal
problem
ID
[8486]
Book
:
Elementary
differential
equations.
By
Earl
D.
Rainville,
Phillip
E.
Bedient.
Macmilliam
Publishing
Co.
NY.
6th
edition.
1981.
Section
:
CHAPTER
16.
Nonlinear
equations.
Section
99.
Clairaut
equation.
EXERCISES
Page
320
Problem
number
:
27
Date
solved
:
Monday, January 27, 2025 at 04:07:13 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} 5 {y^{\prime }}^{2}+3 x y^{\prime }-y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 85
dsolve(5*diff(y(x),x)^2+3*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*}
\frac {c_{1}}{\left (-30 x -10 \sqrt {9 x^{2}+20 y}\right )^{{3}/{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+20 y}}{5} &= 0 \\
\frac {c_{1}}{\left (-30 x +10 \sqrt {9 x^{2}+20 y}\right )^{{3}/{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+20 y}}{5} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 14.354 (sec). Leaf size: 771
DSolve[5*(D[y[x],x])^2+3*x*D[y[x],x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {Root}\left [16 \text {$\#$1}^5+8 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-200000 e^{10 c_1}\&,1\right ] \\
y(x)\to \text {Root}\left [16 \text {$\#$1}^5+8 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-200000 e^{10 c_1}\&,2\right ] \\
y(x)\to \text {Root}\left [16 \text {$\#$1}^5+8 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-200000 e^{10 c_1}\&,3\right ] \\
y(x)\to \text {Root}\left [16 \text {$\#$1}^5+8 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-200000 e^{10 c_1}\&,4\right ] \\
y(x)\to \text {Root}\left [16 \text {$\#$1}^5+8 \text {$\#$1}^4 x^2+\text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-200000 e^{10 c_1}\&,5\right ] \\
y(x)\to \text {Root}\left [3200000 \text {$\#$1}^5+1600000 \text {$\#$1}^4 x^2+200000 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,1\right ] \\
y(x)\to \text {Root}\left [3200000 \text {$\#$1}^5+1600000 \text {$\#$1}^4 x^2+200000 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,2\right ] \\
y(x)\to \text {Root}\left [3200000 \text {$\#$1}^5+1600000 \text {$\#$1}^4 x^2+200000 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,3\right ] \\
y(x)\to \text {Root}\left [3200000 \text {$\#$1}^5+1600000 \text {$\#$1}^4 x^2+200000 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-e^{10 c_1}\&,4\right ] \\
y(x)\to \text {Root}\left [3200000 \text {$\#$1}^5+1600000 \text {$\#$1}^4 x^2+200000 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$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*}