56.4.16 problem 16

Internal problem ID [8905]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 16
Date solved : Monday, January 27, 2025 at 05:19:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 54

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

\[ y = \left (1-\frac {1}{2} x^{3}-\frac {5}{8} x^{4}-\frac {53}{40} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {7}{12} x^{4}+\frac {7}{6} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {53 x^5}{40}-\frac {5 x^4}{8}-\frac {x^3}{2}+1\right )+c_2 \left (\frac {7 x^5}{6}+\frac {7 x^4}{12}+\frac {x^3}{2}+\frac {x^2}{2}+x\right ) \]