83.8.26 problem 27

Internal problem ID [19152]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Misc examples on chapter II at page 25
Problem number : 27
Date solved : Tuesday, January 28, 2025 at 01:10:04 PM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 280

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

\begin{align*} y \left (x \right ) &= \frac {4 a x +\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_{1} \right ) x^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}}{2 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_{1} \right ) x^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {\left (-1-i \sqrt {3}\right ) \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_{1} \right ) x^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}}{4}+\frac {x a \left (i \sqrt {3}-1\right )}{\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_{1} \right ) x^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_{1} \right ) x^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}}{4}-\frac {x \left (1+i \sqrt {3}\right ) a}{\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_{1} \right ) x^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.520 (sec). Leaf size: 325

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

\begin{align*} y(x)\to -\frac {2 a x+\sqrt [3]{2} \left (\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1}} \\ y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1\right ){}^{2/3}+2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) a x}{4 \sqrt [3]{\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1}} \\ y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1\right ){}^{2/3}+2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) a x}{4 \sqrt [3]{\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1}} \\ \end{align*}