56.4.41 problem 38

Internal problem ID [8930]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 38
Date solved : Monday, January 27, 2025 at 05:20:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x +2\right ) y^{\prime \prime }+5 x^{2} y^{\prime }+y \left (1+x \right )&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 48

Order:=6; 
dsolve(2*x^2*(2+x)*diff(y(x), x$2) +5*x^2*diff(y(x),x)+(1+x)*y(x) = 0,y(x),type='series',x=0);
 

\[ y = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {3}{4} x +\frac {15}{32} x^{2}-\frac {35}{128} x^{3}+\frac {315}{2048} x^{4}-\frac {693}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x -\frac {13}{64} x^{2}+\frac {101}{768} x^{3}-\frac {641}{8192} x^{4}+\frac {7303}{163840} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 134

AsymptoticDSolveValue[2*x^2*(2+x)*D[y[x],{x,2}] +5*x^2*D[y[x],x]+(1+x)*y[x] == 0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {693 x^5}{8192}+\frac {315 x^4}{2048}-\frac {35 x^3}{128}+\frac {15 x^2}{32}-\frac {3 x}{4}+1\right )+c_2 \left (\sqrt {x} \left (\frac {7303 x^5}{163840}-\frac {641 x^4}{8192}+\frac {101 x^3}{768}-\frac {13 x^2}{64}+\frac {x}{4}\right )+\sqrt {x} \left (-\frac {693 x^5}{8192}+\frac {315 x^4}{2048}-\frac {35 x^3}{128}+\frac {15 x^2}{32}-\frac {3 x}{4}+1\right ) \log (x)\right ) \]