50.18.4 problem 1(d)

Internal problem ID [8098]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.3. Second-Order Linear Equations: Ordinary Points. Page 169
Problem number : 1(d)
Date solved : Monday, January 27, 2025 at 03:43:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-x^{2} y&=1 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 99

Order:=8; 
dsolve(diff(y(x),x$2)+diff(y(x),x)-x^2*y(x)=1,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {7}{120} x^{5}-\frac {19}{720} x^{6}+\frac {13}{1680} x^{7}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+\frac {13 x^{6}}{720}-\frac {11 x^{7}}{1680}+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 126

AsymptoticDSolveValue[D[y[x],{x,2}]+D[y[x],x]+x^2*y[x]==1,y[x],{x,0,"8"-1}]
 

\[ y(x)\to \frac {31 x^7}{5040}-\frac {11 x^6}{720}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}+c_1 \left (\frac {x^7}{2520}-\frac {x^6}{360}+\frac {x^5}{60}-\frac {x^4}{12}+1\right )+c_2 \left (-\frac {37 x^7}{5040}+\frac {17 x^6}{720}-\frac {x^5}{24}-\frac {x^4}{24}+\frac {x^3}{6}-\frac {x^2}{2}+x\right ) \]