60.1.422 problem 424
Internal
problem
ID
[10436]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
424
Date
solved
:
Monday, January 27, 2025 at 07:43:48 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} x {y^{\prime }}^{2}+a y y^{\prime }+b x&=0 \end{align*}
✓ Solution by Maple
Time used: 0.084 (sec). Leaf size: 217
dsolve(x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)+b*x = 0,y(x), singsol=all)
\begin{align*}
\frac {-c_{1} 2^{\frac {a +2}{2 a +2}} \left (a y-\sqrt {a^{2} y^{2}-4 b \,x^{2}}\right ) {\left (\frac {a \left (-y \left (a +1\right ) \sqrt {a^{2} y^{2}-4 b \,x^{2}}+\left (a^{2}+a \right ) y^{2}-2 b \,x^{2}\right )}{x^{2}}\right )}^{\frac {-a -2}{2 a +2}}+x^{2}}{x} &= 0 \\
\frac {c_{1} 2^{\frac {a +2}{2 a +2}} \left (a y+\sqrt {a^{2} y^{2}-4 b \,x^{2}}\right ) {\left (\frac {a \left (y \left (a +1\right ) \sqrt {a^{2} y^{2}-4 b \,x^{2}}+\left (a^{2}+a \right ) y^{2}-2 b \,x^{2}\right )}{x^{2}}\right )}^{\frac {-a -2}{2 a +2}}+x^{2}}{x} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.099 (sec). Leaf size: 423
DSolve[b*x + a*y[x]*D[y[x],x] + x*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [-\frac {i \left (2 \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 (a+1) \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}-\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}&=c_1-\frac {1}{2} i \log (x),y(x)\right ] \\
\text {Solve}\left [\frac {i \left (2 (a+1) \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (-\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}&=\frac {1}{2} i \log (x)+c_1,y(x)\right ] \\
\end{align*}