50.17.21 problem 4

Internal problem ID [8092]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.2. Series Solutions of First-Order Differential Equations Page 162
Problem number : 4
Date solved : Monday, January 27, 2025 at 03:43:36 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y+1 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 76

Order:=8; 
dsolve(diff(y(x),x)=1+y(x),y(x),type='series',x=0);
 

\[ y = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) y \left (0\right )+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 95

AsymptoticDSolveValue[D[y[x],x]==1+y[x],y[x],{x,0,"8"-1}]
 

\[ y(x)\to \frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+c_1 \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+x \]