Internal problem ID [2427]
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: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {2 x^{2} y^{\prime \prime }+\left (-x^{2}+x \right ) y^{\prime }-y=x^{3}+1} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 57
Order:=6; dsolve(2*x^2*diff(y(x),x$2)+(x-x^2)*diff(y(x),x)-y(x)=1+x^3,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x +\frac {1}{35} x^{2}+\frac {1}{315} x^{3}+\frac {1}{3465} x^{4}+\frac {1}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-1+\frac {1}{14} x^{3}+\frac {1}{126} x^{4}+\frac {1}{1386} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 248
AsymptoticDSolveValue[2*x^2*y''[x]+(x-x^2)*y'[x]-y[x]==1+x^3,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right )}{\sqrt {x}}+c_2 x \left (\frac {x^5}{45045}+\frac {x^4}{3465}+\frac {x^3}{315}+\frac {x^2}{35}+\frac {x}{5}+1\right )+\frac {\left (\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right ) \left (-\frac {6233 x^{11/2}}{1921920}+\frac {2107 x^{9/2}}{95040}-\frac {143 x^{7/2}}{1512}-\frac {x^{5/2}}{140}+\frac {x^{3/2}}{15}-\frac {2 \sqrt {x}}{3}\right )}{\sqrt {x}}+x \left (\frac {x^5}{45045}+\frac {x^4}{3465}+\frac {x^3}{315}+\frac {x^2}{35}+\frac {x}{5}+1\right ) \left (\frac {x^6}{691200}-\frac {2747 x^5}{518918400}+\frac {x^2}{6}-\frac {1}{3 x}\right ) \]