10.6.22 problem 22

Internal problem ID [1239]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Miscellaneous problems, end of chapter 2. Page 133
Problem number : 22
Date solved : Monday, January 27, 2025 at 04:47:09 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {x^{2}-1}{1+y^{2}} \end{align*}

With initial conditions

\begin{align*} y \left (-1\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.116 (sec). Leaf size: 86

dsolve([diff(y(x),x) = (x^2-1)/(1+y(x)^2),y(-1) = 1],y(x), singsol=all)
 

\[ y = \frac {\left (8+4 x^{3}-12 x +4 \sqrt {x^{6}-6 x^{4}+4 x^{3}+9 x^{2}-12 x +8}\right )^{{2}/{3}}-4}{2 \left (8+4 x^{3}-12 x +4 \sqrt {x^{6}-6 x^{4}+4 x^{3}+9 x^{2}-12 x +8}\right )^{{1}/{3}}} \]

✓ Solution by Mathematica

Time used: 2.963 (sec). Leaf size: 97

DSolve[{D[y[x],x]== (x^2-1)/(1+y[x]^2),y[-1]==1},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\sqrt [3]{2} \left (x^3+\sqrt {x^6-6 x^4+4 x^3+9 x^2-12 x+8}-3 x+2\right )^{2/3}-2}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6-6 x^4+4 x^3+9 x^2-12 x+8}-3 x+2}} \]