28.3.7 problem 8

Internal problem ID [7181]
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 : 8
Date solved : Tuesday, September 30, 2025 at 04:25:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 45

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(2*x^2+x)*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1-\frac {4}{5} x +\frac {2}{5} x^{2}-\frac {16}{105} x^{3}+\frac {1}{21} x^{4}-\frac {4}{315} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-144+192 x -96 x^{2}+32 x^{4}-\frac {128}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 208

ode=x^2*D[y[x],{x,2}]+(x+2*x^2)*D[y[x],x]-4*y[x]==2; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (\frac {2 x^2}{3}-\frac {4 x}{3}+1\right )}{x^2}+c_2 \left (-\frac {4 x^5}{315}+\frac {x^4}{21}-\frac {16 x^3}{105}+\frac {2 x^2}{5}-\frac {4 x}{5}+1\right ) x^2+\left (-\frac {4 x^5}{315}+\frac {x^4}{21}-\frac {16 x^3}{105}+\frac {2 x^2}{5}-\frac {4 x}{5}+1\right ) \left (\frac {7 x^6}{2430}+\frac {19 x^5}{2025}+\frac {5 x^4}{216}+\frac {2 x^3}{45}+\frac {x^2}{18}-\frac {1}{4 x^2}-\frac {1}{3 x}\right ) x^2+\frac {\left (\frac {2 x^2}{3}-\frac {4 x}{3}+1\right ) \left (-\frac {x^6}{84}-\frac {4 x^5}{105}-\frac {x^4}{10}-\frac {x^3}{5}-\frac {x^2}{4}\right )}{x^2} \]

✓ Sympy. Time used: 0.302 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (2*x**2 + x)*Derivative(y(x), x) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (- \frac {16 x^{3}}{105} + \frac {2 x^{2}}{5} - \frac {4 x}{5} + 1\right ) + \frac {C_{1} \left (\frac {2 x^{2}}{3} - \frac {4 x}{3} + 1\right )}{x^{2}} + O\left (x^{6}\right ) \]