59.1.216 problem 219

Internal problem ID [9388]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 219
Date solved : Monday, January 27, 2025 at 06:02:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f&=0 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 43

dsolve(diff(f(z),z$2)+2*(z-1)*diff(f(z),z)+4*f(z)=0,f(z), singsol=all)
 

\[ f \left (z \right ) = -\sqrt {\pi }\, \operatorname {erf}\left (i \left (z -1\right )\right ) c_{2} \left (z -1\right ) {\mathrm e}^{-\left (z -1\right )^{2}}+c_{1} {\mathrm e}^{-z \left (z -2\right )} \left (z -1\right )+i c_{2} \]

✓ Solution by Mathematica

Time used: 0.133 (sec). Leaf size: 72

DSolve[D[ f[z],{z,2}]+2*(z-a)*D[ f[z],z]+4*f[z]==0,f[z],z,IncludeSingularSolutions -> True]
 

\[ f(z)\to e^{z (2 a-z)} \left (-\sqrt {\pi } c_2 \sqrt {(a-z)^2} \text {erfi}\left (\sqrt {(a-z)^2}\right )+c_2 e^{(a-z)^2}-2 a c_1+2 c_1 z\right ) \]