13.3 problem 3

Internal problem ID [1244]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number: 3.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]

Solve \begin {gather*} \boxed {\left (-2 x^{2}+1\right ) y^{\prime \prime }+\left (2-6 x \right ) y^{\prime }-2 y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = 0] \end {align*}

With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.002 (sec). Leaf size: 18

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

\[ y \relax (x ) = 1+x^{2}-\frac {2}{3} x^{3}+\frac {11}{6} x^{4}-\frac {9}{5} x^{5}+\mathrm {O}\left (x^{6}\right ) \]

Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 29

AsymptoticDSolveValue[{(1-2*x^2)*y''[x]+(2-6*x)*y'[x]-2*y[x]==0,{y[0]==1,y'[0]==0}},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {9 x^5}{5}+\frac {11 x^4}{6}-\frac {2 x^3}{3}+x^2+1 \]