4.64 problem 61

Internal problem ID [7933]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 61
Date solved : Tuesday, October 22, 2024 at 03:00:10 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

Solve

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

4.64.1 Maple step by step solution

4.64.2 Maple trace

Methods for second order ODEs:
 

4.64.3 Maple dsolve solution

Solving time : 0.019 (sec)
Leaf size : 155

dsolve(x/(1-x)*diff(diff(y(x),x),x)+y(x) = 1/(1-x), 
       y(x),singsol=all)
 

\[ y = -\left (\left (\operatorname {BesselI}\left (0, -x \right )+\operatorname {BesselI}\left (1, -x \right )\right ) \left (\int \frac {-\operatorname {BesselK}\left (0, -x \right )+\operatorname {BesselK}\left (1, -x \right )}{\operatorname {BesselK}\left (1, -x \right ) \operatorname {BesselI}\left (0, x\right ) x^{2}-\operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right ) x^{2}+x +1}d x \right )+\left (\int \frac {\operatorname {BesselI}\left (0, -x \right )+\operatorname {BesselI}\left (1, -x \right )}{\operatorname {BesselK}\left (1, -x \right ) \operatorname {BesselI}\left (0, x\right ) x^{2}-\operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right ) x^{2}+x +1}d x \right ) \left (\operatorname {BesselK}\left (0, -x \right )-\operatorname {BesselK}\left (1, -x \right )\right )+c_1 \operatorname {BesselK}\left (0, -x \right )-c_1 \operatorname {BesselK}\left (1, -x \right )-c_2 \operatorname {BesselI}\left (0, -x \right )-c_2 \operatorname {BesselI}\left (1, -x \right )\right ) x \]

4.64.4 Mathematica DSolve solution

Solving time : 0.248 (sec)
Leaf size : 136

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

\[ y(x)\to e^{-x} x \left (e^x (\operatorname {BesselI}(0,x)-\operatorname {BesselI}(1,x)) \int _1^x2 e^{-K[1]} \sqrt {\pi } \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 K[1]\right )dK[1]-2 \sqrt {\pi } x \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right ) \, _1F_2\left (\frac {1}{2};1,\frac {3}{2};\frac {x^2}{4}\right )+2 \sqrt {\pi } \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right ) \operatorname {BesselI}(0,x)+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right )+c_2 e^x \operatorname {BesselI}(0,x)-c_2 e^x \operatorname {BesselI}(1,x)\right ) \]