73.7.33 problem 33

Internal problem ID [15191]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 8. Review exercises for part of part II. page 143
Problem number : 33
Date solved : Tuesday, January 28, 2025 at 07:41:26 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Bernoulli]

\begin{align*} x y^{3} y^{\prime }&=y^{4}-x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 71

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

\begin{align*} y &= \left (x^{2} \left (c_{1} x^{2}+2\right )\right )^{{1}/{4}} \\ y &= -\left (x^{2} \left (c_{1} x^{2}+2\right )\right )^{{1}/{4}} \\ y &= -i \left (x^{2} \left (c_{1} x^{2}+2\right )\right )^{{1}/{4}} \\ y &= i \left (x^{2} \left (c_{1} x^{2}+2\right )\right )^{{1}/{4}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.551 (sec). Leaf size: 96

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

\begin{align*} y(x)\to -\sqrt {x} \sqrt [4]{2+c_1 x^2} \\ y(x)\to -i \sqrt {x} \sqrt [4]{2+c_1 x^2} \\ y(x)\to i \sqrt {x} \sqrt [4]{2+c_1 x^2} \\ y(x)\to \sqrt {x} \sqrt [4]{2+c_1 x^2} \\ \end{align*}