56.1.31 problem 32

Internal problem ID [8743]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 32
Date solved : Monday, January 27, 2025 at 04:46:06 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 2 t +3 x+\left (x+2\right ) x^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 4.072 (sec). Leaf size: 30

dsolve(2*t+3*x(t)+(x(t)+2)*diff(x(t),t)=0,x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {-\sqrt {4 \left (t -3\right ) c_{1} +1}-1+\left (-4 t +8\right ) c_{1}}{2 c_{1}} \]

✓ Solution by Mathematica

Time used: 60.104 (sec). Leaf size: 1165

DSolve[2*t+3*x[t]+(x[t]+2)*D[x[t],t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -2-\frac {2 (t-3)}{t \sqrt {\frac {3}{(t-3)^2}-\frac {3 (t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-3 \sqrt {\frac {3}{(t-3)^2}-\frac {3 (t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+1} \\ x(t)\to -2+\frac {2 (t-3)}{t \sqrt {\frac {3}{(t-3)^2}-\frac {3 (t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-3 \sqrt {\frac {3}{(t-3)^2}-\frac {3 (t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-1} \\ x(t)\to -2-\frac {2 (t-3)}{t \sqrt {\frac {3}{(t-3)^2}-\frac {3 (t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-3 \sqrt {\frac {3}{(t-3)^2}-\frac {3 (t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+1} \\ x(t)\to -2+\frac {2 (t-3)}{t \sqrt {\frac {3}{(t-3)^2}-\frac {3 (t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-3 \sqrt {\frac {3}{(t-3)^2}-\frac {3 (t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(t-3)^2 \left ((t-3)^2 \cosh \left (\frac {4 c_1}{9}\right )+(t-3)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-1} \\ \end{align*}