60.1.246 problem 247

Internal problem ID [10260]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 247
Date solved : Monday, January 27, 2025 at 06:42:14 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (2+3 x \right ) \left (y-2 x -1\right ) y^{\prime }-y^{2}+y x -7 x^{2}-9 x -3&=0 \end{align*}

✓ Solution by Maple

Time used: 0.576 (sec). Leaf size: 388

dsolve((3*x+2)*(y(x)-2*x-1)*diff(y(x),x)-y(x)^2+x*y(x)-7*x^2-9*x-3=0,y(x), singsol=all)
 

\[ y = \frac {\frac {14 \left (x +\frac {2}{3}\right ) \left (-\frac {4}{1701}+\left (x^{2}+\frac {26}{21} x +\frac {8}{21}\right ) c_{1}^{2}\right ) \left (2 \sqrt {-2187 \left (x +\frac {2}{3}\right )^{2} \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )^{{2}/{3}}}{27}+21 \left (x +\frac {4}{7}\right ) \left (\frac {\left (-\frac {2 \sqrt {-2187 \left (x +\frac {2}{3}\right )^{2} \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) c_{1}^{2}}}{2187}+\left (x +\frac {2}{3}\right ) c_{1} \left (-\frac {2}{729}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right )\right ) \left (1+i \sqrt {3}\right ) \left (2 \sqrt {-2187 \left (x +\frac {2}{3}\right )^{2} \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )^{{1}/{3}}}{9}+\left (x +\frac {2}{3}\right ) \left (-\frac {4 \sqrt {-2187 \left (x +\frac {2}{3}\right )^{2} \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) c_{1}^{2}}}{2187}+\left (x +\frac {2}{3}\right ) \left (\frac {2}{27}+\left (x +\frac {2}{3}\right ) c_{1} \right ) \left (-\frac {2}{27}+\left (x +\frac {2}{3}\right ) c_{1} \right ) c_{1} \right ) \left (i \sqrt {3}-1\right ) c_{1} \right )}{{\left (\frac {\left (1-i \sqrt {3}\right ) \left (2 \sqrt {-2187 \left (x +\frac {2}{3}\right )^{2} \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )^{{2}/{3}}}{81}+\left (x +\frac {2}{3}\right ) c_{1} \left (\frac {2 \left (2 \sqrt {-2187 \left (x +\frac {2}{3}\right )^{2} \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )^{{1}/{3}}}{9}+\left (x +\frac {2}{3}\right ) c_{1} \left (1+i \sqrt {3}\right )\right )\right )}^{2}} \]

✓ Solution by Mathematica

Time used: 6.933 (sec). Leaf size: 127

DSolve[(3*x+2)*(y[x]-2*x-1)*D[y[x],x]-y[x]^2+x*y[x]-7*x^2-9*x-3==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{-\frac {(3 x+2) (11 x-y(x)+7)}{2^{2/3} \sqrt [3]{5} \sqrt [3]{-(3 x+2)^3} (2 x-y(x)+1)}}\frac {1}{K[1]^3+\frac {21 \sqrt [3]{-\frac {1}{2}} K[1]}{2\ 5^{2/3}}+1}dK[1]=\frac {2 \sqrt [3]{2} 5^{2/3} (3 x+2) \log (3 x+2)}{27 \sqrt [3]{-(3 x+2)^3}}+c_1,y(x)\right ] \]