78.12.35 problem 7 (b)

Internal problem ID [18322]
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 17. The Homogeneous Equation with Constant Coefficients. Problems at page 125
Problem number : 7 (b)
Date solved : Tuesday, January 28, 2025 at 08:28:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 x y^{\prime }+x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 56

dsolve(diff(y(x),x$2)+3*x*diff(y(x),x)+x^2*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {x^{2} \left (3+\sqrt {5}\right )}{4}} x \left (\operatorname {KummerM}\left (\frac {3}{4}+\frac {3 \sqrt {5}}{20}, \frac {3}{2}, \frac {\sqrt {5}\, x^{2}}{2}\right ) c_{1} +\operatorname {KummerU}\left (\frac {3}{4}+\frac {3 \sqrt {5}}{20}, \frac {3}{2}, \frac {\sqrt {5}\, x^{2}}{2}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 84

DSolve[D[y[x],{x,2}] +3*x*D[y[x],x]+x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {1}{4} \left (3+\sqrt {5}\right ) x^2} \left (c_1 \operatorname {HermiteH}\left (\frac {1}{10} \left (-5-3 \sqrt {5}\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {1}{20} \left (5+3 \sqrt {5}\right ),\frac {1}{2},\frac {\sqrt {5} x^2}{2}\right )\right ) \]