81.3.19 problem 19

Internal problem ID [18637]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 12:05:11 PM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 60.245 (sec). Leaf size: 1965

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

\begin{align*} y(x)\to \frac {\sqrt {\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {3 x^4+x^2-12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-2 x}-\sqrt {-\frac {12 \sqrt {3} x^2}{\sqrt {\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {3 x^4+x^2-12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-2 x}}-\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {-3 x^4-x^2+12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-4 x}}{2 \sqrt {3}} \\ y(x)\to \frac {\sqrt {\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {3 x^4+x^2-12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-2 x}+\sqrt {-\frac {12 \sqrt {3} x^2}{\sqrt {\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {3 x^4+x^2-12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-2 x}}-\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {-3 x^4-x^2+12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-4 x}}{2 \sqrt {3}} \\ y(x)\to -\frac {\sqrt {\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {3 x^4+x^2-12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-2 x}+\sqrt {\frac {12 \sqrt {3} x^2}{\sqrt {\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {3 x^4+x^2-12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-2 x}}-\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {-3 x^4-x^2+12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-4 x}}{2 \sqrt {3}} \\ y(x)\to \frac {\sqrt {\frac {12 \sqrt {3} x^2}{\sqrt {\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {3 x^4+x^2-12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-2 x}}-\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {-3 x^4-x^2+12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-4 x}-\sqrt {\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}+\frac {3 x^4+x^2-12 c_1}{\sqrt [3]{-9 x^5+54 x^4+x^3+\sqrt {\left (-9 x^5+54 x^4+x^3+36 c_1 x\right ){}^2-\left (3 x^4+x^2-12 c_1\right ){}^3}+36 c_1 x}}-2 x}}{2 \sqrt {3}} \\ \end{align*}