60.1.474 problem 477
Internal
problem
ID
[10488]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
477
Date
solved
:
Monday, January 27, 2025 at 07:55:19 PM
CAS
classification
:
[[_homogeneous, `class C`], _rational, _dAlembert]
\begin{align*} a y {y^{\prime }}^{2}+\left (2 x -b \right ) y^{\prime }-y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.210 (sec). Leaf size: 741
dsolve(a*y(x)*diff(y(x),x)^2+(2*x-b)*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*}
y &= -\frac {-2 x +b}{2 \sqrt {-a}} \\
y &= \frac {-2 x +b}{2 \sqrt {-a}} \\
y &= 0 \\
-4 a \left (\int _{}^{y}\frac {16 \left (\frac {1}{16}+\left (\left (\frac {b}{4}-\frac {x}{2}\right ) \sqrt {4 \textit {\_f}^{2} a +b^{2}-4 b x +4 x^{2}}+\textit {\_f}^{2} a +\frac {b^{2}}{4}-b x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {-2 \textit {\_f}^{2} a +2 \sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} -\sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b -4 \textit {\_a}^{2}+4 b \textit {\_a} -b^{2}}{\left (-4 \textit {\_f}^{2} a +2 \sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} -\sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b -4 \textit {\_a}^{2}+4 b \textit {\_a} -b^{2}\right )^{2} \sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}}d \textit {\_a} \right )\right ) \textit {\_f}}{\left (-2 x +b \right ) \sqrt {4 \textit {\_f}^{2} a +b^{2}-4 b x +4 x^{2}}+4 \textit {\_f}^{2} a +b^{2}-4 b x +4 x^{2}}d \textit {\_f} \right )+2 \left (\int _{\textit {\_b}}^{x}\frac {-2 \textit {\_a} +b +\sqrt {4 a y^{2}+\left (-2 \textit {\_a} +b \right )^{2}}}{\left (-2 \textit {\_a} +b \right ) \sqrt {4 a y^{2}+\left (-2 \textit {\_a} +b \right )^{2}}+4 a y^{2}+\left (-2 \textit {\_a} +b \right )^{2}}d \textit {\_a} \right )+c_{1} &= 0 \\
-4 a \left (\int _{}^{y}\frac {16 \left (\frac {1}{16}+\left (\left (-\frac {b}{4}+\frac {x}{2}\right ) \sqrt {4 \textit {\_f}^{2} a +b^{2}-4 b x +4 x^{2}}+\textit {\_f}^{2} a +\frac {b^{2}}{4}-b x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {2 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}+2 \sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} -\sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b}{\left (4 \textit {\_f}^{2} a +2 \sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} -\sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}\right )^{2} \sqrt {4 \textit {\_f}^{2} a +4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}}d \textit {\_a} \right )\right ) \textit {\_f}}{\left (2 x -b \right ) \sqrt {4 \textit {\_f}^{2} a +b^{2}-4 b x +4 x^{2}}+4 \textit {\_f}^{2} a +b^{2}-4 b x +4 x^{2}}d \textit {\_f} \right )-2 \left (\int _{\textit {\_b}}^{x}\frac {2 \textit {\_a} -b +\sqrt {4 a y^{2}+\left (-2 \textit {\_a} +b \right )^{2}}}{\left (2 \textit {\_a} -b \right ) \sqrt {4 a y^{2}+\left (-2 \textit {\_a} +b \right )^{2}}+4 a y^{2}+\left (-2 \textit {\_a} +b \right )^{2}}d \textit {\_a} \right )+c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.861 (sec). Leaf size: 187
DSolve[-y[x] + (-b + 2*x)*D[y[x],x] + a*y[x]*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}+b-2 x} \\
y(x)\to \sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}+b-2 x} \\
y(x)\to -\frac {e^{\frac {c_1}{2}} \sqrt {-2 b+4 x+e^{c_1}}}{2 \sqrt {a}} \\
y(x)\to \frac {e^{\frac {c_1}{2}} \sqrt {-2 b+4 x+e^{c_1}}}{2 \sqrt {a}} \\
y(x)\to 0 \\
y(x)\to -\frac {i (b-2 x)}{2 \sqrt {a}} \\
y(x)\to \frac {i (b-2 x)}{2 \sqrt {a}} \\
\end{align*}