Internal problem ID [4215]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XV. page 194
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x}-\sqrt {x}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✗ Solution by Maple
Order:=6; dsolve(2*x^2*diff(y(x),x$2)-(3*x+2)*diff(y(x),x)+(2*x-1)/x*y(x)=x^(1/2),y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.225 (sec). Leaf size: 222
AsymptoticDSolveValue[2*x^2*y''[x]-(3*x+2)*y'[x]+(2*x-1)/x*y[x]==x^(1/2),y[x],{x,0,5}]
\[ y(x)\to \frac {1}{256} e^{-1/x} \left (-\frac {405405 x^5}{16}+\frac {45045 x^4}{16}-\frac {693 x^3}{2}+\frac {189 x^2}{4}-7 x+1\right ) x^4 \left (\frac {2 e^{\frac {1}{x}} \left (15663375 x^7+20072325 x^6+10329540 x^5+4131816 x^4+2754544 x^3+5509088 x^2-64 x-64\right )}{x^{3/2}}-11018112 \sqrt {\pi } \text {Erfi}\left (\frac {1}{\sqrt {x}}\right )\right )+c_2 e^{-1/x} \left (-\frac {405405 x^5}{16}+\frac {45045 x^4}{16}-\frac {693 x^3}{2}+\frac {189 x^2}{4}-7 x+1\right ) x^4+\frac {\left (\frac {5 x}{2}+1\right ) \left (-\frac {15015 x^6}{64}+\frac {693 x^5}{20}-\frac {189 x^4}{32}+\frac {7 x^3}{6}-\frac {x^2}{4}\right )}{\sqrt {x}}+\frac {c_1 \left (\frac {5 x}{2}+1\right )}{\sqrt {x}} \]