29.26.16 problem 752
Internal
problem
ID
[5337]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
26
Problem
number
:
752
Date
solved
:
Monday, January 27, 2025 at 11:15:45 AM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} {y^{\prime }}^{2}+x^{2}&=4 y \end{align*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 136
dsolve(diff(y(x),x)^2+x^2 = 4*y(x),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )^{2}} \\
y \left (x \right ) &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (-\frac {c_{1} \sqrt {2}\, x}{2}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {c_{1} \sqrt {2}\, x}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (-\frac {c_{1} \sqrt {2}\, x}{2}\right )^{2}} \\
y \left (x \right ) &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {c_{1} \sqrt {2}\, x}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {c_{1} \sqrt {2}\, x}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {c_{1} \sqrt {2}\, x}{2}\right )^{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 17.688 (sec). Leaf size: 152
DSolve[(D[y[x],x])^2+x^2==4*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [-\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )+\frac {x \sqrt {4 y(x)-x^2}}{2 x^2-4 y(x)}+\int \frac {x \left (x^2-4 y(x)\right )}{\left (x^2-2 y(x)\right )^2} \, dx&=c_1,y(x)\right ] \\
\text {Solve}\left [\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )-\frac {x \sqrt {4 y(x)-x^2}}{2 \left (x^2-2 y(x)\right )}+\int \frac {x \left (x^2-4 y(x)\right )}{\left (x^2-2 y(x)\right )^2} \, dx&=c_1,y(x)\right ] \\
\end{align*}