60.1.427 problem 429

Internal problem ID [10441]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 429
Date solved : Monday, January 27, 2025 at 07:43:54 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.099 (sec). Leaf size: 72

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

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

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 90

DSolve[b*y[x] - (-a - b + 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 {a+b}{-b+a c_1}\right ) \\ y(x)\to \frac {-2 \sqrt {a^2 b x (a+b)}+a^2+a b (x+1)}{a^2} \\ y(x)\to \frac {2 \sqrt {a^2 b x (a+b)}+a^2+a b (x+1)}{a^2} \\ \end{align*}