34.4.2 problem 2

Internal problem ID [6056]
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 : 2
Date solved : Monday, January 27, 2025 at 01:34:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+y^{\prime }+p x y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 44

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} p \,x^{2}+\frac {1}{64} p^{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {p}{4} x^{2}-\frac {3}{128} p^{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 72

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

\[ y(x)\to c_1 \left (\frac {p^2 x^4}{64}-\frac {p x^2}{4}+1\right )+c_2 \left (-\frac {3}{128} p^2 x^4+\left (\frac {p^2 x^4}{64}-\frac {p x^2}{4}+1\right ) \log (x)+\frac {p x^2}{4}\right ) \]