29.32.13 problem 947
Internal
problem
ID
[5525]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
32
Problem
number
:
947
Date
solved
:
Monday, January 27, 2025 at 11:35:21 AM
CAS
classification
:
[[_homogeneous, `class C`], _rational, _dAlembert]
\begin{align*} y {y^{\prime }}^{2}-\left (-2 b x +a \right ) y^{\prime }-b y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.226 (sec). Leaf size: 149
dsolve(y(x)*diff(y(x),x)^2-(-2*b*x+a)*diff(y(x),x)-b*y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {-2 b x +a}{2 \sqrt {-b}} \\
y \left (x \right ) &= \frac {-2 b x +a}{2 \sqrt {-b}} \\
y \left (x \right ) &= 0 \\
y \left (x \right ) &= \sqrt {\frac {c_{1} b +\sqrt {c_{1} b \left (-2 b x +a \right )^{2}}}{b}} \\
y \left (x \right ) &= -\sqrt {\frac {c_{1} b +\sqrt {c_{1} b \left (-2 b x +a \right )^{2}}}{b}} \\
y \left (x \right ) &= \sqrt {-\frac {-c_{1} b +\sqrt {c_{1} b \left (-2 b x +a \right )^{2}}}{b}} \\
y \left (x \right ) &= -\sqrt {\frac {c_{1} b -\sqrt {c_{1} b \left (-2 b x +a \right )^{2}}}{b}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.114 (sec). Leaf size: 409
DSolve[y[x] (D[y[x],x])^2-(a-2 b x)D[y[x],x]-b y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {\left (b-\sqrt {b^2}\right ) \log (y(x))}{b}-\frac {-b \log \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}-a-2 \sqrt {b^2} x\right )+\sqrt {b^2} \log \left (b \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}-a-2 \sqrt {b^2} x\right )\right )-\left (\sqrt {b^2}+b\right ) \log \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}+a-2 \sqrt {b^2} x\right )}{2 \sqrt {b^2}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {-b \log \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}-a-2 \sqrt {b^2} x\right )+\sqrt {b^2} \log \left (b \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}-a-2 \sqrt {b^2} x\right )\right )-\left (\sqrt {b^2}+b\right ) \log \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}+a-2 \sqrt {b^2} x\right )}{2 \sqrt {b^2}}+\frac {\left (\sqrt {b^2}+b\right ) \log (y(x))}{b}&=c_1,y(x)\right ] \\
y(x)\to -\frac {i (2 b x-a)}{2 \sqrt {b}} \\
y(x)\to \frac {i (2 b x-a)}{2 \sqrt {b}} \\
\end{align*}