82.42.2 problem Ex. 2
Internal
problem
ID
[18958]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
VIII.
Exact
differential
equations,
and
equations
of
particular
forms.
Integration
in
series.
problems
at
page
97
Problem
number
:
Ex.
2
Date
solved
:
Tuesday, January 28, 2025 at 12:38:26 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{\prime \prime }&=\frac {1}{\sqrt {a y}} \end{align*}
✓ Solution by Maple
Time used: 0.043 (sec). Leaf size: 79
dsolve(diff(y(x),x$2)=1/sqrt(a*y(x)),y(x), singsol=all)
\begin{align*}
\int _{}^{y \left (x \right )}\frac {\sqrt {\textit {\_a} a}}{\sqrt {\sqrt {\textit {\_a} a}\, \sqrt {\textit {\_a}}\, \left (c_{1} +4 \sqrt {\textit {\_a}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\
-\int _{}^{y \left (x \right )}\frac {\sqrt {\textit {\_a} a}}{\sqrt {\sqrt {\textit {\_a} a}\, \sqrt {\textit {\_a}}\, \left (c_{1} +4 \sqrt {\textit {\_a}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.134 (sec). Leaf size: 1570
DSolve[D[y[x],{x,2}]==1/Sqrt[a*y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {a^4 c_1{}^4+288 a^2 c_1 x^2+576 a^2 c_2 c_1 x+288 a^2 c_2{}^2 c_1+3 a^2 c_1{}^2 \sqrt [3]{-a^6 c_1{}^6+720 a^4 c_1{}^3 x^2+1440 a^4 c_2 c_1{}^3 x+720 a^4 c_2{}^2 c_1{}^3+10368 a^2 x^4+41472 a^2 c_2 x^3+62208 a^2 c_2{}^2 x^2+41472 a^2 c_2{}^3 x+10368 a^2 c_2{}^4+48 \sqrt {-a^4 (x+c_2){}^2 \left (a^2 c_1{}^3-36 x^2-72 c_2 x-36 c_2{}^2\right ){}^3}}+\left (-a^6 c_1{}^6+720 a^4 c_1{}^3 x^2+1440 a^4 c_2 c_1{}^3 x+720 a^4 c_2{}^2 c_1{}^3+10368 a^2 x^4+41472 a^2 c_2 x^3+62208 a^2 c_2{}^2 x^2+41472 a^2 c_2{}^3 x+10368 a^2 c_2{}^4+48 \sqrt {-a^4 (x+c_2){}^2 \left (a^2 c_1{}^3-36 x^2-72 c_2 x-36 c_2{}^2\right ){}^3}\right ){}^{2/3}}{16 a \sqrt [3]{-a^6 c_1{}^6+720 a^4 c_1{}^3 x^2+1440 a^4 c_2 c_1{}^3 x+720 a^4 c_2{}^2 c_1{}^3+10368 a^2 x^4+41472 a^2 c_2 x^3+62208 a^2 c_2{}^2 x^2+41472 a^2 c_2{}^3 x+10368 a^2 c_2{}^4+48 \sqrt {-a^4 (x+c_2){}^2 \left (a^2 c_1{}^3-36 x^2-72 c_2 x-36 c_2{}^2\right ){}^3}}} \\
y(x)\to \frac {\left (-1-i \sqrt {3}\right ) a^4 c_1{}^4+6 a^2 c_1 \left (c_1 \sqrt [3]{-a^6 c_1{}^6+720 a^4 c_1{}^3 x^2+1440 a^4 c_2 c_1{}^3 x+720 a^4 c_2{}^2 c_1{}^3+10368 a^2 x^4+41472 a^2 c_2 x^3+62208 a^2 c_2{}^2 x^2+41472 a^2 c_2{}^3 x+10368 a^2 c_2{}^4+48 \sqrt {a^4 (x+c_2){}^2 \left (-a^2 c_1{}^3+36 x^2+72 c_2 x+36 c_2{}^2\right ){}^3}}-48 i \left (\sqrt {3}-i\right ) x^2-96 i \left (\sqrt {3}-i\right ) c_2 x-48 i \sqrt {3} c_2{}^2-48 c_2{}^2\right )+i \left (\sqrt {3}+i\right ) \left (-a^6 c_1{}^6+720 a^4 c_1{}^3 x^2+1440 a^4 c_2 c_1{}^3 x+720 a^4 c_2{}^2 c_1{}^3+10368 a^2 x^4+41472 a^2 c_2 x^3+62208 a^2 c_2{}^2 x^2+41472 a^2 c_2{}^3 x+10368 a^2 c_2{}^4+48 \sqrt {-a^4 (x+c_2){}^2 \left (a^2 c_1{}^3-36 x^2-72 c_2 x-36 c_2{}^2\right ){}^3}\right ){}^{2/3}}{32 a \sqrt [3]{-a^6 c_1{}^6+720 a^4 c_1{}^3 x^2+1440 a^4 c_2 c_1{}^3 x+720 a^4 c_2{}^2 c_1{}^3+10368 a^2 x^4+41472 a^2 c_2 x^3+62208 a^2 c_2{}^2 x^2+41472 a^2 c_2{}^3 x+10368 a^2 c_2{}^4+48 \sqrt {-a^4 (x+c_2){}^2 \left (a^2 c_1{}^3-36 x^2-72 c_2 x-36 c_2{}^2\right ){}^3}}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) a^4 c_1{}^4+6 a^2 c_1 \left (c_1 \sqrt [3]{-a^6 c_1{}^6+720 a^4 c_1{}^3 x^2+1440 a^4 c_2 c_1{}^3 x+720 a^4 c_2{}^2 c_1{}^3+10368 a^2 x^4+41472 a^2 c_2 x^3+62208 a^2 c_2{}^2 x^2+41472 a^2 c_2{}^3 x+10368 a^2 c_2{}^4+48 \sqrt {a^4 (x+c_2){}^2 \left (-a^2 c_1{}^3+36 x^2+72 c_2 x+36 c_2{}^2\right ){}^3}}+48 i \left (\sqrt {3}+i\right ) x^2+96 i \left (\sqrt {3}+i\right ) c_2 x+48 i \sqrt {3} c_2{}^2-48 c_2{}^2\right )-i \left (\sqrt {3}-i\right ) \left (-a^6 c_1{}^6+720 a^4 c_1{}^3 x^2+1440 a^4 c_2 c_1{}^3 x+720 a^4 c_2{}^2 c_1{}^3+10368 a^2 x^4+41472 a^2 c_2 x^3+62208 a^2 c_2{}^2 x^2+41472 a^2 c_2{}^3 x+10368 a^2 c_2{}^4+48 \sqrt {-a^4 (x+c_2){}^2 \left (a^2 c_1{}^3-36 x^2-72 c_2 x-36 c_2{}^2\right ){}^3}\right ){}^{2/3}}{32 a \sqrt [3]{-a^6 c_1{}^6+720 a^4 c_1{}^3 x^2+1440 a^4 c_2 c_1{}^3 x+720 a^4 c_2{}^2 c_1{}^3+10368 a^2 x^4+41472 a^2 c_2 x^3+62208 a^2 c_2{}^2 x^2+41472 a^2 c_2{}^3 x+10368 a^2 c_2{}^4+48 \sqrt {-a^4 (x+c_2){}^2 \left (a^2 c_1{}^3-36 x^2-72 c_2 x-36 c_2{}^2\right ){}^3}}} \\
\end{align*}