83.22.20 problem 20

Internal problem ID [19274]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 20
Date solved : Tuesday, January 28, 2025 at 01:19:13 PM
CAS classification : [_rational]

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

✗ Solution by Maple

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

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 3.656 (sec). Leaf size: 150

DSolve[(a*D[y[x],x]^2-b)*x*y[x]+(b*x^2-a*y[x]^2+c)*D[y[x],x]==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*}