60.1.426 problem 428

Internal problem ID [10440]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 428
Date solved : Monday, January 27, 2025 at 07:43:53 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.098 (sec). Leaf size: 66

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

\begin{align*} y &= \frac {-b x +c -2 \sqrt {-b c x}}{a} \\ y &= \frac {-b x +c +2 \sqrt {-b c x}}{a} \\ y &= \frac {c_{1} \left (a c_{1} x +b x +c \right )}{a c_{1} +b} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 80

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

\begin{align*} y(x)\to c_1 \left (x+\frac {c}{b+a c_1}\right ) \\ y(x)\to \frac {\left (\sqrt {c}-i \sqrt {b} \sqrt {x}\right )^2}{a} \\ y(x)\to \frac {\left (\sqrt {c}+i \sqrt {b} \sqrt {x}\right )^2}{a} \\ \end{align*}