2.21 problem 22

Internal problem ID [8110]
Book : Second order enumerated odes
Section : section 2
Problem number : 22
Date solved : Tuesday, October 22, 2024 at 03:00:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

Solve

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+y&=4 \cos \left (\ln \left (1+x \right )\right ) \end{align*}

2.21.1 Maple step by step solution

2.21.2 Maple trace

Methods for second order ODEs:
 

2.21.3 Maple dsolve solution

Solving time : 0.019 (sec)
Leaf size : 280

dsolve((x^2+1)*diff(diff(y(x),x),x)+(1+x)*diff(y(x),x)+y(x) = 4*cos(ln(1+x)), 
       y(x),singsol=all)
 

\[ y = \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right ) c_2 +\left (x +i\right )^{\frac {1}{2}-\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right ) c_1 -80 \left (\int \frac {\left (i x -1\right ) \cos \left (\ln \left (1+x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )}{\left (x^{2}+1\right ) \left (10 \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right ) \left (\left (-1-i+\left (-1+i\right ) x \right ) \operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )+\left (1+i\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )\right )+\operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}-\frac {3 i}{2}, \frac {3}{2}+\frac {i}{2}\right ], \left [\frac {5}{2}-\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right ) \left (1+7 i+\left (7-i\right ) x \right )\right )}d x \right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )-80 \left (\int \frac {\operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right ) \cos \left (\ln \left (1+x \right )\right ) \left (x +i\right )^{\frac {1}{2}+\frac {i}{2}}}{7 \left (\frac {10 \left (\left (1-i+\left (-1-i\right ) x \right ) \operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )+\left (-1+i\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )}{7}+\operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right ) \left (-1+\frac {i}{7}+\left (\frac {1}{7}+i\right ) x \right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}-\frac {3 i}{2}, \frac {3}{2}+\frac {i}{2}\right ], \left [\frac {5}{2}-\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )\right ) \left (x^{2}+1\right )}d x \right ) \left (x +i\right )^{\frac {1}{2}-\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right ) \]

2.21.4 Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

DSolve[{(1+x^2)*D[y[x],{x,2}]+(1+x)*D[y[x],x]+y[x]==4*Cos[Log[1+x]],{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

Not solved