78.10.10 problem 11 (b)

Internal problem ID [18271]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 15. The General Solution of the Homogeneous Equation. Problems at page 117
Problem number : 11 (b)
Date solved : Tuesday, January 28, 2025 at 11:44:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 16

dsolve(diff(y(x),x$2)+2*x*diff(y(x),x)+(1+x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {x^{2}}{2}} \left (c_{2} x +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 77

DSolve[D[y[x],{x,2}] +2*D[y[x],x]+(1+x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} e^{-x} \left (2 c_1 e^{-\frac {i x^2}{2}} \operatorname {HermiteH}\left (-\frac {1}{2},\sqrt [4]{-1} x\right )+\sqrt {2} c_2 \sqrt [4]{i x^2} \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselI}\left (-\frac {1}{4},\frac {i x^2}{2}\right )\right ) \]