Internal problem ID [1197]
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: 18.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {\left (2-x \right ) y^{\prime \prime }+2 y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = a_{0}, y^{\prime }\relax (0) = a_{1}] \end {align*}
With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 40
Order:=6; dsolve([(2-x)*diff(y(x),x$2)+2*y(x)=0,y(0) = a__0, D(y)(0) = a__1],y(x),type='series',x=0);
\[ y \relax (x ) = a_{0} +a_{1} x -\frac {1}{2} a_{0} x^{2}+\left (-\frac {a_{1}}{6}-\frac {a_{0}}{12}\right ) x^{3}+\left (\frac {a_{0}}{48}-\frac {a_{1}}{24}\right ) x^{4}+\left (-\frac {a_{1}}{240}+\frac {a_{0}}{96}\right ) x^{5}+\mathrm {O}\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 79
AsymptoticDSolveValue[{(2-x)*y''[x]+2*y[x]==0,{y[0]==a0,y'[0]==a1}},y[x],{x,0,5}]
\[ y(x)\to \frac {1}{20} x^5 \left (\frac {1}{6} \left (\frac {\text {a0}}{2}+\text {a1}\right )+\frac {\text {a0}}{8}-\frac {\text {a1}}{4}\right )+\frac {1}{12} x^4 \left (\frac {\text {a0}}{4}-\frac {\text {a1}}{2}\right )+\frac {1}{6} x^3 \left (-\frac {\text {a0}}{2}-\text {a1}\right )-\frac {\text {a0} x^2}{2}+\text {a0}+\text {a1} x \]