60.1.381 problem 382

Internal problem ID [10395]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 382
Date solved : Monday, January 27, 2025 at 07:38:57 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 174

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

\begin{align*} y &= -\frac {4 c \ln \left (\sqrt {a^{2}+4 b}\, x +\sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}\right )+\sqrt {a^{2}+4 b}\, \left (a \,x^{2}+x \sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}-4 c_{1} \right )}{4 \sqrt {a^{2}+4 b}} \\ y &= -\frac {-4 c \ln \left (\sqrt {a^{2}+4 b}\, x +\sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}\right )+\sqrt {a^{2}+4 b}\, \left (a \,x^{2}-x \sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}-4 c_{1} \right )}{4 \sqrt {a^{2}+4 b}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.490 (sec). Leaf size: 213

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

\begin{align*} y(x)\to -\frac {c^{3/2} \sqrt {\frac {x^2 \left (a^2+4 b\right )}{c}+4} \arcsin \left (\frac {x \sqrt {-a^2-4 b}}{2 \sqrt {c}}\right )}{\sqrt {-a^2-4 b} \sqrt {x^2 \left (a^2+4 b\right )+4 c}}-\frac {1}{4} x \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}+a x\right )+c_1 \\ y(x)\to \frac {1}{4} x \sqrt {x^2 \left (a^2+4 b\right )+4 c}+\frac {c \log \left (\sqrt {a^2+4 b} \sqrt {x^2 \left (a^2+4 b\right )+4 c}+a^2 x+4 b x\right )}{\sqrt {a^2+4 b}}-\frac {a x^2}{4}+c_1 \\ \end{align*}