34.4.13 problem 13

Internal problem ID [6067]
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
Date solved : Monday, January 27, 2025 at 01:34:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x}&=\sqrt {x} \end{align*}

Using series method with expansion around

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

✗ 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: 0.227 (sec). Leaf size: 237

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

\[ y(x)\to -\frac {1}{128} i e^{-1/x} \left (-\frac {405405 x^5}{16}+\frac {45045 x^4}{16}-\frac {693 x^3}{2}+\frac {189 x^2}{4}-7 x+1\right ) x^3 \left (-5509056 i \sqrt {\pi } x \text {erfi}\left (\frac {1}{\sqrt {x}}\right )+\frac {i e^{\frac {1}{x}} \left (15663375 x^7+20072325 x^6+10329540 x^5+4131816 x^4+2754544 x^3+5509088 x^2-64 x-64\right )}{\sqrt {x}}+5509056 \sqrt {\pi } x\right )+c_2 e^{-1/x} \left (-\frac {405405 x^5}{16}+\frac {45045 x^4}{16}-\frac {693 x^3}{2}+\frac {189 x^2}{4}-7 x+1\right ) x^4+\frac {\left (\frac {5 x}{2}+1\right ) \left (-\frac {15015 x^6}{64}+\frac {693 x^5}{20}-\frac {189 x^4}{32}+\frac {7 x^3}{6}-\frac {x^2}{4}\right )}{\sqrt {x}}+\frac {c_1 \left (\frac {5 x}{2}+1\right )}{\sqrt {x}} \]