34.4.5 problem 5

Internal problem ID [6059]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 5
Date solved : Monday, January 27, 2025 at 01:34:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 64

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

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {4}{3} x +\frac {5}{6} x^{2}-\frac {1}{3} x^{3}+\frac {7}{72} x^{4}-\frac {1}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (6 x^{2}-8 x^{3}+5 x^{4}-2 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-4 x +5 x^{2}+2 x^{3}-4 x^{4}+\frac {7}{3} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {13 x^4-12 x^3-4 x^2+8 x+4}{4 x}-\frac {1}{2} x \left (5 x^2-8 x+6\right ) \log (x)\right )+c_2 \left (\frac {7 x^5}{72}-\frac {x^4}{3}+\frac {5 x^3}{6}-\frac {4 x^2}{3}+x\right ) \]