34.3.4 problem 4

Internal problem ID [6043]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XIV. page 177
Problem number : 4
Date solved : Monday, January 27, 2025 at 01:34:19 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+a^{3} x^{2} y&=2 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 50

Order:=6; 
dsolve(x*diff(y(x),x$2)+2*diff(y(x),x)+a^3*x^2*y(x)=2,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {1}{12} a^{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-\frac {1}{6} a^{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x}+x \left (1-\frac {1}{20} a^{3} x^{3}+\operatorname {O}\left (x^{5}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 136

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

\[ y(x)\to c_1 \left (\frac {a^6 x^6}{504}-\frac {a^3 x^3}{12}+1\right )+\frac {c_2 \left (\frac {a^6 x^6}{180}-\frac {a^3 x^3}{6}+1\right )}{x}+\left (2 x-\frac {a^3 x^4}{12}\right ) \left (\frac {a^6 x^6}{504}-\frac {a^3 x^3}{12}+1\right )+\frac {\left (\frac {a^3 x^5}{30}-x^2\right ) \left (\frac {a^6 x^6}{180}-\frac {a^3 x^3}{6}+1\right )}{x} \]