80.4.3 problem 3

Internal problem ID [18554]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 31. Problems at page 85
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 11:58:11 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+\frac {y}{x}&=\frac {\sin \left (x \right )}{y^{3}} \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 156

dsolve(diff(y(x),x)+y(x)/x=sin(x)/y(x)^3,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {{\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{{1}/{4}}}{x} \\ y \left (x \right ) &= -\frac {{\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{{1}/{4}}}{x} \\ y \left (x \right ) &= -\frac {i {\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{{1}/{4}}}{x} \\ y \left (x \right ) &= \frac {i {\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{{1}/{4}}}{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.476 (sec). Leaf size: 114

DSolve[D[y[x],x]+y[x]/x==Sin[x]/y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{9 \left (x^2-2\right ) \sin (x)-3 x \left (x^2-6\right ) \cos (x)+c_1}}{x} \\ y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{9 \left (x^2-2\right ) \sin (x)-3 x \left (x^2-6\right ) \cos (x)+c_1}}{x} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{9 \left (x^2-2\right ) \sin (x)-3 x \left (x^2-6\right ) \cos (x)+c_1}}{x} \\ \end{align*}