41.1.4 problem Ex. 6(iii), page 257

Internal problem ID [6817]
Book : A treatise on Differential Equations by A. R. Forsyth. 6th edition. 1929. Macmillan Co. ltd. New York, reprinted 1956
Section : Chapter VI. Note I. Integration of linear equations in series by the method of Frobenius. page 243
Problem number : Ex. 6(iii), page 257
Date solved : Tuesday, January 28, 2025 at 03:10:35 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} \left (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-\left (12 x^{2}+4\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 46

Order:=6; 
dsolve(x^3*(1+x^2)*diff(y(x),x$3)-(2+4*x^2)*x^2*diff(y(x),x$2)+(4+10*x^2)*x*diff(y(x),x)-(4+12*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = x \left (c_3 \left (2+2 x^{2}+\operatorname {O}\left (x^{6}\right )\right )+x \left (\left (1+\operatorname {O}\left (x^{6}\right )\right ) c_1 +c_2 \left (\ln \left (x \right ) \left (2+\operatorname {O}\left (x^{6}\right )\right )+\left (5+\operatorname {O}\left (x^{6}\right )\right )\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 30

AsymptoticDSolveValue[x^3*(1+x^2)*D[y[x],{x,3}]-(2+4*x^2)*x^2*D[y[x],{x,2}]+(4+10*x^2)*x*D[y[x],x]-(4+12*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (2 x^3+2 x\right )+c_2 x^2+c_3 x^2 \log (x) \]