44.2.22 problem 22

Internal problem ID [6954]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Section 1.2 Initial value problems. Exercises 1.2 at page 19
Problem number : 22
Date solved : Monday, January 27, 2025 at 02:38:16 PM
CAS classification : [_separable]

\begin{align*} \left (1+y^{3}\right ) y^{\prime }&=x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 20

dsolve((1+y(x)^3)*diff(y(x),x)=x^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 60.113 (sec). Leaf size: 871

DSolve[(1+y[x]^3)*D[y[x],x]==x^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {\frac {-4 x^3+\left (27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {\frac {8 \left (x^3+3 c_1\right )}{3 \sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}-\frac {2}{3} \sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}+\frac {4 \sqrt {6}}{\sqrt {\frac {-4 x^3+\left (27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}}}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {8 \left (x^3+3 c_1\right )}{3 \sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}-\frac {2}{3} \sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}+\frac {4 \sqrt {6}}{\sqrt {\frac {-4 x^3+\left (27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}}}}-\frac {\sqrt {\frac {-4 x^3+\left (27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}}}{\sqrt {6}} \\ y(x)\to \frac {\sqrt {\frac {-4 x^3+\left (27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {\frac {8 \left (x^3+3 c_1\right )}{3 \sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}-\frac {2}{3} \sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}-\frac {4 \sqrt {6}}{\sqrt {\frac {-4 x^3+\left (27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}}}} \\ y(x)\to \frac {\sqrt {\frac {-4 x^3+\left (27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}}}{\sqrt {6}}+\frac {1}{2} \sqrt {\frac {8 \left (x^3+3 c_1\right )}{3 \sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}-\frac {2}{3} \sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}-\frac {4 \sqrt {6}}{\sqrt {\frac {-4 x^3+\left (27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27+\sqrt {729+64 \left (x^3+3 c_1\right ){}^3}}}}}} \\ \end{align*}