Internal problem ID [1198]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number: 19.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x +1\right ) y^{\prime \prime }+2 \left (x -1\right )^{2} y^{\prime }+3 y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (1) = a_{0}, y^{\prime }\relax (1) = a_{1}] \end {align*}
With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 40
Order:=6; dsolve([(1+x)*diff(y(x),x$2)+2*(x-1)^2*diff(y(x),x)+3*y(x)=0,y(1) = a__0, D(y)(1) = a__1],y(x),type='series',x=1);
\[ y \relax (x ) = a_{0} +a_{1} \left (x -1\right )-\frac {3}{4} a_{0} \left (x -1\right )^{2}+\left (\frac {a_{0}}{8}-\frac {a_{1}}{4}\right ) \left (x -1\right )^{3}+\left (\frac {a_{0}}{16}-\frac {a_{1}}{48}\right ) \left (x -1\right )^{4}+\left (\frac {3 a_{0}}{64}+\frac {a_{1}}{40}\right ) \left (x -1\right )^{5}+\mathrm {O}\left (\left (x -1\right )^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 95
AsymptoticDSolveValue[{(1+x)*y''[x]+2*(x-1)^2*y'[x]+3*y[x]==0,{y[1]==a0,y'[1]==a1}},y[x],{x,1,5}]
\[ y(x)\to \frac {1}{20} (x-1)^5 \left (\frac {1}{4} \left (\frac {3 \text {a1}}{2}-\frac {3 \text {a0}}{4}\right )+\frac {9 \text {a0}}{8}+\frac {\text {a1}}{8}\right )+\frac {1}{12} (x-1)^4 \left (\frac {3 \text {a0}}{4}-\frac {\text {a1}}{4}\right )+\frac {1}{6} (x-1)^3 \left (\frac {3 \text {a0}}{4}-\frac {3 \text {a1}}{2}\right )-\frac {3}{4} \text {a0} (x-1)^2+\text {a0}+\text {a1} (x-1) \]