15.25.4 problem 3

Internal problem ID [3391]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 43, page 209
Problem number : 3
Date solved : Monday, January 27, 2025 at 07:35:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 54

Order:=6; 
dsolve(x*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=x^3,y(x),type='series',x=0);
 

\[ y \left (x \right ) = x^{4} \left (\frac {1}{12}+\frac {1}{80} x +\operatorname {O}\left (x^{2}\right )\right )+\ln \left (x \right ) \left (-x +\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1+x -\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{72} x^{4}-\frac {1}{480} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.078 (sec). Leaf size: 139

AsymptoticDSolveValue[x*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==x^3,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {x^4}{72}-\frac {x^3}{12}-\frac {x^2}{2}-x \log (x)+1\right )+x \left (\frac {1}{36} x^6 (2-3 \log (x))+\frac {1}{25} x^5 (5 \log (x)-1)+\frac {1}{16} x^4 (-4 \log (x)-3)+\frac {x^3}{3}\right )+\left (-\frac {x^6}{12}+\frac {x^5}{5}-\frac {x^4}{4}\right ) \left (-\frac {x^4}{72}-\frac {x^3}{12}-\frac {x^2}{2}-x \log (x)+1\right )+c_2 x \]