83.7.6 problem 4 (b)

Internal problem ID [19124]
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. Exercise II (F) at page 24
Problem number : 4 (b)
Date solved : Tuesday, January 28, 2025 at 12:58:47 PM
CAS classification : [_rational]

\begin{align*} \left (20 x^{2}+8 y x +4 y^{2}+3 y^{3}\right ) y+4 \left (x^{2}+y x +y^{2}+y^{3}\right ) x y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 38

dsolve((20*x^2+8*x*y(x)+4*y(x)^2+3*y(x)^3)*y(x)+4*(x^2+x*y(x)+y(x)^2+y(x)^3)*x*diff(y(x),x)=0,y(x), singsol=all)
 

\[ 4 x^{5} y \left (x \right )+2 x^{4} y \left (x \right )^{2}+\frac {4 x^{3} y \left (x \right )^{3}}{3}+y \left (x \right )^{4} x^{3}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.199 (sec). Leaf size: 2173

DSolve[(20*x^2+8*x*y[x]+4*y[x]^2+3*y[x]^3)*y[x]+4*(x^2+x*y[x]+y[x]^2+y[x]^3)*x*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{6} \left (-\sqrt {-\frac {18 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}+\frac {6 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{x^3}-12 x+4}-3 \sqrt {\frac {2 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}-\frac {2 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{3 x^3}+\frac {8 \left (27 x^2-9 x+2\right )}{9 \sqrt {-\frac {18 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}+\frac {6 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{x^3}-12 x+4}}-\frac {8 x}{3}+\frac {8}{9}}-2\right ) \\ y(x)\to \frac {1}{6} \left (-\sqrt {-\frac {18 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}+\frac {6 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{x^3}-12 x+4}+3 \sqrt {\frac {2 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}-\frac {2 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{3 x^3}+\frac {8 \left (27 x^2-9 x+2\right )}{9 \sqrt {-\frac {18 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}+\frac {6 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{x^3}-12 x+4}}-\frac {8 x}{3}+\frac {8}{9}}-2\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {-\frac {18 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}+\frac {6 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{x^3}-12 x+4}-3 \sqrt {\frac {2 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}-\frac {2 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{3 x^3}-\frac {8 \left (27 x^2-9 x+2\right )}{9 \sqrt {-\frac {18 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}+\frac {6 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{x^3}-12 x+4}}-\frac {8 x}{3}+\frac {8}{9}}-2\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {-\frac {18 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}+\frac {6 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{x^3}-12 x+4}+3 \sqrt {\frac {2 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}-\frac {2 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{3 x^3}-\frac {8 \left (27 x^2-9 x+2\right )}{9 \sqrt {-\frac {18 \left (x^5+c_1\right )}{\sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}+\frac {6 \sqrt [3]{27 x^{13}-5 x^{12}+9 c_1 x^7-3 c_1 x^6+\sqrt {x^9 \left (27 \left (x^5+c_1\right ){}^3+x^3 \left (-27 x^7+5 x^6-9 c_1 x+3 c_1\right ){}^2\right )}}}{x^3}-12 x+4}}-\frac {8 x}{3}+\frac {8}{9}}-2\right ) \\ \end{align*}