60.1.168 problem 169

Internal problem ID [10182]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 169
Date solved : Tuesday, January 28, 2025 at 04:26:30 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: 0.003 (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)
 

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

✓ Solution by Mathematica

Time used: 1.339 (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]
 

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