15.25.13 problem 12

Internal problem ID [3400]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 43, page 209
Problem number : 12
Date solved : Monday, January 27, 2025 at 07:35:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 40

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

\[ y \left (x \right ) = c_{1} x^{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-2-2 x -\frac {1}{2} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+x^{3} \left (\frac {2}{3}+\frac {1}{6} x +\operatorname {O}\left (x^{3}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.310 (sec). Leaf size: 39

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

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