60.1.482 problem 485

Internal problem ID [10496]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 485
Date solved : Monday, January 27, 2025 at 08:16:50 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.355 (sec). Leaf size: 1218

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

\begin{align*} y &= \frac {\sqrt {a \left (b \,x^{2}-c +2 x \sqrt {-b c}\right )}}{a} \\ y &= \frac {\sqrt {a \left (b \,x^{2}-2 x \sqrt {-b c}-c \right )}}{a} \\ y &= -\frac {\sqrt {a \left (b \,x^{2}-c +2 x \sqrt {-b c}\right )}}{a} \\ y &= -\frac {\sqrt {a \left (b \,x^{2}-2 x \sqrt {-b c}-c \right )}}{a} \\ y &= 0 \\ -2 a^{2} \left (\int _{}^{y}\frac {4 \left (\frac {1}{4}+b \left (\left (-\textit {\_f}^{2} a +b \,x^{2}+c \right ) \sqrt {a^{2} \textit {\_f}^{4}-2 a \left (b \,x^{2}-c \right ) \textit {\_f}^{2}+\left (b \,x^{2}+c \right )^{2}}+x^{4} b^{2}-2 x^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (\left (b \,\textit {\_a}^{2}-\textit {\_f}^{2} a +c \right ) \sqrt {\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}}+\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +a^{2} \textit {\_f}^{4}+c^{2}\right ) \textit {\_a}}{\sqrt {\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}}\, {\left (\left (b \,\textit {\_a}^{2}-\textit {\_f}^{2} a +c \right ) \sqrt {\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}}+\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}\right )}^{2}}d \textit {\_a} \right )\right ) \textit {\_f}}{\left (-\textit {\_f}^{2} a +b \,x^{2}+c \right ) \sqrt {a^{2} \textit {\_f}^{4}-2 a \left (b \,x^{2}-c \right ) \textit {\_f}^{2}+\left (b \,x^{2}+c \right )^{2}}+x^{4} b^{2}-2 x^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}}d \textit {\_f} \right )-a \left (\int _{\textit {\_b}}^{x}\frac {-a y^{2}-b \,\textit {\_a}^{2}-c -\sqrt {y^{4} a^{2}-2 a \left (b \,\textit {\_a}^{2}-c \right ) y^{2}+\left (b \,\textit {\_a}^{2}+c \right )^{2}}}{\left (\left (-a y^{2}+b \,\textit {\_a}^{2}+c \right ) \sqrt {y^{4} a^{2}-2 a \left (b \,\textit {\_a}^{2}-c \right ) y^{2}+\left (b \,\textit {\_a}^{2}+c \right )^{2}}+y^{4} a^{2}-2 a \left (b \,\textit {\_a}^{2}-c \right ) y^{2}+\left (b \,\textit {\_a}^{2}+c \right )^{2}\right ) \textit {\_a}}d \textit {\_a} \right )+c_{1} &= 0 \\ -2 a^{2} \left (\int _{}^{y}\frac {4 \textit {\_f} \left (\frac {1}{4}+b \left (\left (\textit {\_f}^{2} a -b \,x^{2}-c \right ) \sqrt {a^{2} \textit {\_f}^{4}-2 a \left (b \,x^{2}-c \right ) \textit {\_f}^{2}+\left (b \,x^{2}+c \right )^{2}}+x^{4} b^{2}-2 x^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}\right ) \left (\int _{\textit {\_b}}^{x}-\frac {\textit {\_a} \left (\left (-b \,\textit {\_a}^{2}+\textit {\_f}^{2} a -c \right ) \sqrt {\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}}+\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +a^{2} \textit {\_f}^{4}+c^{2}\right )}{\sqrt {\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}}\, {\left (\left (-b \,\textit {\_a}^{2}+\textit {\_f}^{2} a -c \right ) \sqrt {\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}}+\textit {\_a}^{4} b^{2}-2 \textit {\_a}^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}\right )}^{2}}d \textit {\_a} \right )\right )}{\left (\textit {\_f}^{2} a -b \,x^{2}-c \right ) \sqrt {a^{2} \textit {\_f}^{4}-2 a \left (b \,x^{2}-c \right ) \textit {\_f}^{2}+\left (b \,x^{2}+c \right )^{2}}+x^{4} b^{2}-2 x^{2} \left (\textit {\_f}^{2} a -c \right ) b +\left (\textit {\_f}^{2} a +c \right )^{2}}d \textit {\_f} \right )-a \left (\int _{\textit {\_b}}^{x}\frac {-a y^{2}-b \,\textit {\_a}^{2}+\sqrt {y^{4} a^{2}-2 a \left (b \,\textit {\_a}^{2}-c \right ) y^{2}+\left (b \,\textit {\_a}^{2}+c \right )^{2}}-c}{\left (\left (a y^{2}-b \,\textit {\_a}^{2}-c \right ) \sqrt {y^{4} a^{2}-2 a \left (b \,\textit {\_a}^{2}-c \right ) y^{2}+\left (b \,\textit {\_a}^{2}+c \right )^{2}}+y^{4} a^{2}-2 a \left (b \,\textit {\_a}^{2}-c \right ) y^{2}+\left (b \,\textit {\_a}^{2}+c \right )^{2}\right ) \textit {\_a}}d \textit {\_a} \right )+c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.650 (sec). Leaf size: 155

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

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