10.2.8 problem 8
Internal
problem
ID
[1136]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Section
2.2.
Page
48
Problem
number
:
8
Date
solved
:
Monday, January 27, 2025 at 04:35:18 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {x^{2}}{1+y^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 264
dsolve(diff(y(x),x) = x^2/(1+y(x)^2),y(x), singsol=all)
\begin{align*}
y &= \frac {\left (4 x^{3}+12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}+4}\right )^{{2}/{3}}-4}{2 \left (4 x^{3}+12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}+4}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 x^{3}+12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}+4}\right )^{{2}/{3}}+4 i \sqrt {3}-4}{4 \left (4 x^{3}+12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}+4}\right )^{{1}/{3}}} \\
y &= \frac {i \left (4 x^{3}+12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}+4}\right )^{{2}/{3}} \sqrt {3}+4 i \sqrt {3}-\left (4 x^{3}+12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}+4}\right )^{{2}/{3}}+4}{4 \left (4 x^{3}+12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}+4}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.213 (sec). Leaf size: 307
DSolve[D[y[x],x]== x^2/(1+y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {-2+\sqrt [3]{2} \left (x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}} \\
y(x)\to \frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\
\end{align*}