60.1.64 problem 64

Internal problem ID [10078]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 64
Date solved : Monday, January 27, 2025 at 06:23:57 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }-\sqrt {\frac {a y^{2}+b y+c}{a \,x^{2}+b x +c}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 153

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

\[ -\frac {\sqrt {a \,x^{2}+b x +c}\, \left (-\ln \left (2\right )+\ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{\sqrt {a}}\right )\right ) \sqrt {\frac {a y^{2}+b y+c}{a \,x^{2}+b x +c}}-\left (c_{1} \sqrt {a}-\ln \left (2\right )+\ln \left (\frac {2 \sqrt {a y^{2}+b y+c}\, \sqrt {a}+2 a y+b}{\sqrt {a}}\right )\right ) \sqrt {a y^{2}+b y+c}}{\sqrt {a}\, \sqrt {a y^{2}+b y+c}} = 0 \]

✓ Solution by Mathematica

Time used: 5.601 (sec). Leaf size: 142

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

\begin{align*} y(x)\to \frac {e^{-\sqrt {a} c_1} \left (2 \sqrt {a} \left (-1+e^{2 \sqrt {a} c_1}\right ) \sqrt {x (a x+b)+c}+b \left (-1+e^{\sqrt {a} c_1}\right ){}^2+2 a x \left (1+e^{2 \sqrt {a} c_1}\right )\right )}{4 a} \\ y(x)\to -\frac {\sqrt {b^2-4 a c}+b}{2 a} \\ y(x)\to \frac {\sqrt {b^2-4 a c}-b}{2 a} \\ \end{align*}