33.1.6 problem problem 169

Internal problem ID [6024]
Book : Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section : Abel ODE with constant invariant
Problem number : problem 169
Date solved : Tuesday, January 28, 2025 at 03:10:12 PM
CAS classification : [_rational, _Abel]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 126

dsolve((a*x+b)^2*diff(y(x),x)+(a*x+b)*y(x)^3+c*y(x)^2 = 0,y(x), singsol=all)
 

\[ \frac {\left (\sqrt {a}\, b +a^{{3}/{2}} x \right ) {\mathrm e}^{-\frac {\left (\left (-a x -b +c \right ) y+a \left (a x +b \right )\right ) \left (\left (a x +b +c \right ) y+a \left (a x +b \right )\right )}{2 y^{2} \left (a x +b \right )^{2} a}}+\frac {c \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {1}{2 a}} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (c y+a \left (a x +b \right )\right )}{2 \sqrt {a}\, y \left (a x +b \right )}\right )}{2}+c_1 \,a^{{3}/{2}}}{a^{{3}/{2}}} = 0 \]

✓ Solution by Mathematica

Time used: 1.356 (sec). Leaf size: 149

DSolve[(a*x+b)^2*D[y[x],x]+(a*x+b)*y[x]^3+c*y[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [-\frac {c}{\sqrt {-a (a x+b)^2}}=\frac {2 \exp \left (\frac {1}{2} \left (-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}\right )^2\right )}{\sqrt {2 \pi } \text {erfi}\left (\frac {-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}}{\sqrt {2}}\right )+2 c_1},y(x)\right ] \]