83.22.18 problem 18

Internal problem ID [19272]
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 : 18
Date solved : Tuesday, January 28, 2025 at 01:19:10 PM
CAS classification : [[_homogeneous, `class G`], _rational, _dAlembert]

\begin{align*} 2 y&=x y^{\prime }+\frac {a}{y^{\prime }} \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 788

dsolve(2*y(x)=x*diff(y(x),x)+a/diff(y(x),x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {x \left (\frac {4 x^{2}}{\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}}+2 x +\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}\right )}{12 c_{1}}+\frac {3 a c_{1} \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}}{\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}+2 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}+4 x^{2}} \\ y \left (x \right ) &= -\frac {x \left (\left (1+i \sqrt {3}\right ) \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}-4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}-4 \left (i \sqrt {3}-1\right ) x^{2}\right )}{24 \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}} c_{1}}+\frac {6 a c_{1} \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}}{4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}-4 x^{2}+4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}-\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}} \\ y \left (x \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}+4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}-4 x^{2} \left (1+i \sqrt {3}\right )\right ) x}{24 \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}} c_{1}}-\frac {6 a c_{1} \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}}{-i \sqrt {3}\, \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}+4 i \sqrt {3}\, x^{2}+\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}-4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}+4 x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.158 (sec). Leaf size: 1553

DSolve[2*y[x]==x*D[y[x],x]+a/D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (a^4 x^4+\left (-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}\right ){}^{2/3}-a^2 x^2 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}+8 a e^{3 c_1} x\right )}{4 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}} \\ y(x)\to \frac {i e^{-\frac {3 c_1}{2}} \left (-\left (\left (\sqrt {3}-i\right ) a^4 x^4\right )+\left (\sqrt {3}+i\right ) \left (-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}\right ){}^{2/3}+2 i a^2 x^2 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}-8 \left (\sqrt {3}-i\right ) a e^{3 c_1} x\right )}{8 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}} \\ y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (i \left (\sqrt {3}+i\right ) a^4 x^4-i \left (\sqrt {3}-i\right ) \left (-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}\right ){}^{2/3}-2 a^2 x^2 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}+8 i \left (\sqrt {3}+i\right ) a e^{3 c_1} x\right )}{8 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}} \\ y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (a^4 x^4+\left (a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}+a^2 x^2 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}+8 a e^{3 c_1} x\right )}{4 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (\left (-1-i \sqrt {3}\right ) a^4 x^4+i \left (\sqrt {3}+i\right ) \left (a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}+2 a^2 x^2 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}-8 i \left (\sqrt {3}-i\right ) a e^{3 c_1} x\right )}{8 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (i \left (\sqrt {3}+i\right ) a^4 x^4-i \left (\sqrt {3}-i\right ) \left (a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}+2 a^2 x^2 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}+8 i \left (\sqrt {3}+i\right ) a e^{3 c_1} x\right )}{8 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ \end{align*}