60.1.375 problem 376
Internal
problem
ID
[10389]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
376
Date
solved
:
Monday, January 27, 2025 at 07:38:52 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2}+a y^{\prime }+b y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 245
dsolve(diff(y(x),x)^2+a*diff(y(x),x)+b*y(x) = 0,y(x), singsol=all)
\begin{align*}
y &= -\frac {a^{2} \left (\operatorname {LambertW}\left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (-x +c_{1} \right ) b -a}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (-x +c_{1} \right ) b -a}{a}}}{a}\right )}{4 b} \\
y &= -\frac {a^{2} \left (\operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (-x +c_{1} \right ) b -a}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (-x +c_{1} \right ) b -a}{a}}}{a}\right )}{4 b} \\
y &= {\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {\left (-x +c_{1} \right ) b -a}{a}}}{a \sqrt {-\frac {1}{b}}}\right )-a +\left (-x +c_{1} \right ) b}{a}} \left (a \sqrt {-\frac {1}{b}}+{\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {\left (-x +c_{1} \right ) b -a}{a}}}{a \sqrt {-\frac {1}{b}}}\right )-a +\left (-x +c_{1} \right ) b}{a}}\right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.817 (sec). Leaf size: 119
DSolve[b*y[x] + a*D[y[x],x] + D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {$\#$1} b}+a \log \left (b \left (\sqrt {a^2-4 \text {$\#$1} b}-a\right )\right )}{2 b}\&\right ]\left [\frac {x}{2}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {$\#$1} b}-a \log \left (b \left (\sqrt {a^2-4 \text {$\#$1} b}+a\right )\right )}{2 b}\&\right ]\left [-\frac {x}{2}+c_1\right ] \\
y(x)\to 0 \\
\end{align*}