45.1.22 problem 34

Internal problem ID [7222]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.1.2 page 230
Problem number : 34
Date solved : Monday, January 27, 2025 at 02:48:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+{\mathrm e}^{x} y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 54

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

\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 63

AsymptoticDSolveValue[D[y[x],{x,2}]+Exp[x]*D[y[x],x]-y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{120}-\frac {x^3}{6}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^5}{120}-\frac {x^4}{24}+\frac {x^3}{6}-\frac {x^2}{2}+x\right ) \]