Internal problem ID [12383]
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).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_linear]
\[ \boxed {y^{\prime }-\frac {y}{-x^{2}+1}=\sqrt {x}} \] With initial conditions \begin {align*} \left [y \left (\frac {1}{2}\right ) = 1\right ] \end {align*}
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 145
dsolve([diff(y(x),x)=y(x)/(1-x^2)+sqrt(x),y(1/2) = 1],y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (4 i \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )-12 i \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )+\sqrt {3}\, \sqrt {2}-2 \sqrt {3}\right ) \left (x +1\right )}{6 \sqrt {-x^{2}+1}}+\frac {-2 \sqrt {x +1}\, \sqrt {-2 x +2}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {x +1}\, \sqrt {-2 x +2}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )+2 x^{3}-2 x}{\sqrt {x}\, \left (3 x -3\right )} \]
✓ Solution by Mathematica
Time used: 1.562 (sec). Leaf size: 215
DSolve[{y'[x]==y[x]/(1-x^2)+Sqrt[x],{y[1/2]==1}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {4 \sqrt {1-x^2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^2\right )-4 \sqrt {1-x^2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^2\right )-\sqrt {2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {1}{4}\right ) \sqrt {-((x-1) x)} \sqrt {x+1}+2 \sqrt {2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {1}{4}\right ) \sqrt {-((x-1) x)} \sqrt {x+1}-4 x^3+4 x-\sqrt {6} \sqrt {-((x-1) x)} \sqrt {x+1}+2 \sqrt {3} \sqrt {-((x-1) x)} \sqrt {x+1}}{6 \sqrt {1-x} \sqrt {-((x-1) x)}} \]