82.40.1 problem Ex. 1

Internal problem ID [18947]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 94
Problem number : Ex. 1
Date solved : Tuesday, January 28, 2025 at 12:38:07 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

\begin{align*} x y^{\prime \prime \prime }+\left (x^{2}-3\right ) y^{\prime \prime }+4 x y^{\prime }+2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 79

dsolve(x*diff(y(x),x$3)+(x^2-3)*diff(y(x),x$2)+4*x*diff(y(x),x)+2*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{2}}{2}} x^{5} c_{1} \operatorname {Ei}_{1}\left (-\frac {x^{2}}{2}\right )-\frac {\left (i \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) c_{2} \sqrt {2}\, \sqrt {\pi }-30 c_3 \right ) x^{5} {\mathrm e}^{-\frac {x^{2}}{2}}}{30}-\frac {c_{2} x^{4}}{15}+2 c_{1} x^{3}-\frac {c_{2} x^{2}}{15}+4 c_{1} x -\frac {c_{2}}{5} \]

✓ Solution by Mathematica

Time used: 0.147 (sec). Leaf size: 103

DSolve[x*D[y[x],{x,3}]+(x^2-3)*D[y[x],{x,2}]+4*x*D[y[x],x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{240} e^{-\frac {x^2}{2}} \left (8 c_2 \left (\sqrt {2 \pi } x^5 \text {erfi}\left (\frac {x}{\sqrt {2}}\right )-2 e^{\frac {x^2}{2}} \left (x^4+x^2+3\right )\right )+15 c_3 x \left (x^4 \operatorname {ExpIntegralEi}\left (\frac {x^2}{2}\right )-2 e^{\frac {x^2}{2}} \left (x^2+2\right )\right )+240 c_1 x^5\right ) \]