81.3.32 problem 32

Internal problem ID [18650]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 32
Date solved : Tuesday, January 28, 2025 at 12:08:39 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} \left (x -3 y+4\right ) y^{\prime }&=5 x -7 y \end{align*}

✓ Solution by Maple

Time used: 1.769 (sec). Leaf size: 28

dsolve((x-3*y(x)+4)*diff(y(x),x)=5*x-7*y(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {-1+\sqrt {1+\left (-48 x +168\right ) c_{1}}+60 \left (x -2\right ) c_{1}}{36 c_{1}} \]

✓ Solution by Mathematica

Time used: 60.146 (sec). Leaf size: 1339

DSolve[(x-3*y[x]+4)*D[y[x],x]==5*x-7*y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{3} \left (x+\frac {2 (2 x-7)}{2 x \sqrt {\frac {3}{(7-2 x)^2}-\frac {3 (7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^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 )}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-7 \sqrt {\frac {3}{(7-2 x)^2}-\frac {3 (7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^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 )}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+1}+4\right ) \\ y(x)\to \frac {1}{3} \left (x+\frac {14-4 x}{2 x \sqrt {\frac {3}{(7-2 x)^2}-\frac {3 (7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^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 )}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-7 \sqrt {\frac {3}{(7-2 x)^2}-\frac {3 (7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^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 )}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-1}+4\right ) \\ y(x)\to \frac {1}{3} \left (x+\frac {2 (2 x-7)}{2 x \sqrt {\frac {3}{(7-2 x)^2}-\frac {3 (7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^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 )}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-7 \sqrt {\frac {3}{(7-2 x)^2}-\frac {3 (7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^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 )}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}+1}+4\right ) \\ y(x)\to \frac {1}{3} \left (x+\frac {14-4 x}{2 x \sqrt {\frac {3}{(7-2 x)^2}-\frac {3 (7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^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 )}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-7 \sqrt {\frac {3}{(7-2 x)^2}-\frac {3 (7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+3 (7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+2}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^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 )}{(7-2 x)^2 \left ((7-2 x)^2 \cosh \left (\frac {4 c_1}{9}\right )+(7-2 x)^2 \sinh \left (\frac {4 c_1}{9}\right )+1\right ){}^2}}}-1}+4\right ) \\ \end{align*}