29.18.16 problem 492

Internal problem ID [5090]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 18
Problem number : 492
Date solved : Monday, January 27, 2025 at 10:10:23 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} \left (5-x +6 y\right ) y^{\prime }&=3-x +4 y \end{align*}

✓ Solution by Maple

Time used: 3.842 (sec). Leaf size: 29

dsolve((5-x+6*y(x))*diff(y(x),x) = 3-x+4*y(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\sqrt {9+\left (-8 x -8\right ) c_{1}}-3+\left (4 x -4\right ) c_{1}}{8 c_{1}} \]

✓ Solution by Mathematica

Time used: 60.169 (sec). Leaf size: 1177

DSolve[(5-x+6 y[x])D[y[x],x]==3-x+4 y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{6} \left (x+\frac {2 (x+1)}{x \sqrt {\frac {3}{(x+1)^2}-\frac {3 (x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^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 )}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+\sqrt {\frac {3}{(x+1)^2}-\frac {3 (x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^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 )}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+1}-5\right ) \\ y(x)\to \frac {1}{6} \left (x-\frac {2 (x+1)}{x \sqrt {\frac {3}{(x+1)^2}-\frac {3 (x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^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 )}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+\sqrt {\frac {3}{(x+1)^2}-\frac {3 (x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^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 )}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-1}-5\right ) \\ y(x)\to \frac {1}{6} \left (x+\frac {2 (x+1)}{x \sqrt {\frac {3}{(x+1)^2}-\frac {3 (x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^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 )}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+\sqrt {\frac {3}{(x+1)^2}-\frac {3 (x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^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 )}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+1}-5\right ) \\ y(x)\to \frac {1}{6} \left (x-\frac {2 (x+1)}{x \sqrt {\frac {3}{(x+1)^2}-\frac {3 (x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^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 )}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+\sqrt {\frac {3}{(x+1)^2}-\frac {3 (x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^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 )}{(x+1)^2 \left ((x+1)^2 \cosh \left (\frac {4 c_1}{9}\right )+(x+1)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-1}-5\right ) \\ \end{align*}