71.8.7 problem 4 (c)

Internal problem ID [14441]
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 (c)
Date solved : Tuesday, January 28, 2025 at 06:31:13 AM
CAS classification : [_linear]

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

With initial conditions

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

✓ Solution by Maple

Time used: 1.982 (sec). Leaf size: 141

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

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

✓ Solution by Mathematica

Time used: 0.099 (sec). Leaf size: 63

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

\[ y(x)\to \exp \left (\int _2^x\frac {1}{1-K[1]^2}dK[1]\right ) \left (\int _2^x\exp \left (-\int _2^{K[2]}\frac {1}{1-K[1]^2}dK[1]\right ) \sqrt {K[2]}dK[2]+1\right ) \]