73.6.12 problem 7.5 (b)

Internal problem ID [15151]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 7. The exact form and general integrating fators. Additional exercises. page 141
Problem number : 7.5 (b)
Date solved : Tuesday, January 28, 2025 at 07:38:12 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 16

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

\[ y^{4}-y^{3} c_{1} +x = 0 \]

✓ Solution by Mathematica

Time used: 52.298 (sec). Leaf size: 1270

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

\begin{align*} y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}+\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{4}}-\frac {1}{2} \sqrt {-\frac {c_1{}^3}{4 \sqrt {\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}+\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{4}}}-\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}-\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{2}}+\frac {c_1}{4} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}+\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{4}}+\frac {1}{2} \sqrt {-\frac {c_1{}^3}{4 \sqrt {\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}+\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{4}}}-\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}-\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{2}}+\frac {c_1}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}+\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{4}}-\frac {1}{2} \sqrt {\frac {c_1{}^3}{4 \sqrt {\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}+\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{4}}}-\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}-\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{2}}+\frac {c_1}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}+\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{4}}+\frac {1}{2} \sqrt {\frac {c_1{}^3}{4 \sqrt {\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}+\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{4}}}-\frac {\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}{\sqrt [3]{2} 3^{2/3}}-\frac {4 \sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {-768 x^3+81 c_1{}^4 x^2}+9 c_1{}^2 x}}+\frac {c_1{}^2}{2}}+\frac {c_1}{4} \\ y(x)\to 0 \\ \end{align*}