40.6.7 problem 16
Internal
problem
ID
[6687]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
10.
Singular
solutions,
Extraneous
loci.
Supplemetary
problems.
Page
74
Problem
number
:
16
Date
solved
:
Monday, January 27, 2025 at 02:22:25 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} 2 y&={y^{\prime }}^{2}+4 x y^{\prime } \end{align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 692
dsolve(2*y(x)=diff(y(x),x)^2+4*x*diff(y(x),x),y(x), singsol=all)
\begin{align*}
y &= \frac {\left (\left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}-2 x \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}+4 x^{2}\right ) \left (\left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}+6 x \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}+4 x^{2}\right )}{8 \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}} \\
y &= \frac {\left (i \sqrt {3}\, \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}-4 i \sqrt {3}\, x^{2}+\left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}+4 x \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}+4 x^{2}\right ) \left (i \sqrt {3}\, \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}-4 i \sqrt {3}\, x^{2}+\left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}-12 x \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}+4 x^{2}\right )}{32 \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}} \\
y &= \frac {\left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}+4 x^{2}+4 x \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}+\left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}+4 x^{2}-12 x \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}+\left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}\right )}{32 \left (12 c_1 -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{2}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.239 (sec). Leaf size: 1344
DSolve[2*y[x]==D[y[x],x]^2+4*x*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{2} \left (-x^2+\frac {x \left (x^3+2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}+\sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\
y(x)\to \frac {1}{4} \left (-2 x^2-\frac {\left (1+i \sqrt {3}\right ) x \left (x^3+2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\
y(x)\to \frac {1}{4} \left (-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3+2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\
y(x)\to \frac {1}{2} \left (-x^2+\sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}+\frac {x^4-2 \sqrt {2} e^{3 c_1} x}{\sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}\right ) \\
y(x)\to \frac {1}{4} \left (-2 x^2+\frac {\left (1+i \sqrt {3}\right ) x \left (-x^3+2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\
y(x)\to \frac {1}{4} \left (-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3-2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\
\end{align*}