15.25.3 problem 2

Internal problem ID [3390]
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 : 2
Date solved : Monday, January 27, 2025 at 07:35:31 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 44

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

\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{3} \left (\frac {1}{9}+\frac {2}{225} x^{2}+\operatorname {O}\left (x^{3}\right )\right )+\left (-\frac {1}{2} x^{2}-\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.053 (sec). Leaf size: 188

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

\[ y(x)\to c_2 \left (\frac {x^6}{288}+\frac {x^4}{16}+\frac {x^2}{2}+1\right )+c_1 \left (x^6 \left (\frac {\log (x)}{144}-\frac {1}{108}\right )+x^4 \left (\frac {\log (x)}{8}-\frac {1}{8}\right )+x^2 \left (\log (x)-\frac {1}{2}\right )+2 \log (x)+1\right )+\left (\frac {x^6}{288}+\frac {x^4}{16}+\frac {x^2}{2}+1\right ) \left (\frac {1}{100} x^5 (7-10 \log (x))+\frac {1}{18} x^3 (-6 \log (x)-1)\right )+\left (\frac {x^5}{20}+\frac {x^3}{6}\right ) \left (x^6 \left (\frac {\log (x)}{144}-\frac {1}{108}\right )+x^4 \left (\frac {\log (x)}{8}-\frac {1}{8}\right )+x^2 \left (\log (x)-\frac {1}{2}\right )+2 \log (x)+1\right ) \]