71.8.5 problem 4 (a)

Internal problem ID [14439]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.4.4, page 115
Problem number : 4 (a)
Date solved : Tuesday, January 28, 2025 at 06:31:08 AM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (\frac {1}{2}\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 4.258 (sec). Leaf size: 139

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

\[ y = \frac {\left (12 i \operatorname {EllipticE}\left (\frac {\sqrt {6}}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-8 i \operatorname {EllipticF}\left (\frac {\sqrt {3}}{2}, \sqrt {2}\right )-\sqrt {2}\, \sqrt {3}+2 \sqrt {3}\right ) \left (x +1\right )}{6 \sqrt {-x^{2}+1}}+\frac {-2 \sqrt {x +1}\, \sqrt {-2 x +2}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {x +1}\, \sqrt {-2 x +2}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )+2 x^{3}-2 x}{\sqrt {x}\, \left (3 x -3\right )} \]

✓ Solution by Mathematica

Time used: 0.123 (sec). Leaf size: 69

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

\[ y(x)\to \exp \left (\int _{\frac {1}{2}}^x\frac {1}{1-K[1]^2}dK[1]\right ) \left (\int _{\frac {1}{2}}^x\exp \left (-\int _{\frac {1}{2}}^{K[2]}\frac {1}{1-K[1]^2}dK[1]\right ) \sqrt {K[2]}dK[2]+1\right ) \]