15.23.18 problem 22

Internal problem ID [3368]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 22
Date solved : Monday, January 27, 2025 at 07:35:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 45

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

\[ y \left (x \right ) = \frac {c_{1} \left (1-x -\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {29}{1440} x^{4}-\frac {71}{50400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x -\frac {2}{35} x^{2}-\frac {16}{945} x^{3}+\frac {73}{20790} x^{4}+\frac {1481}{1351350} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 84

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

\[ y(x)\to c_1 x \left (\frac {1481 x^5}{1351350}+\frac {73 x^4}{20790}-\frac {16 x^3}{945}-\frac {2 x^2}{35}+\frac {x}{5}+1\right )+\frac {c_2 \left (-\frac {71 x^5}{50400}+\frac {29 x^4}{1440}+\frac {x^3}{36}-\frac {x^2}{4}-x+1\right )}{\sqrt {x}} \]