40.16.4 problem 11

Internal problem ID [6795]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 25. Integration in series. Supplemetary problems. Page 205
Problem number : 11
Date solved : Wednesday, March 05, 2025 at 02:46:01 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }&=2 x^{2}+3 y \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.000 (sec). Leaf size: 52

Order:=6; 
ode:=diff(y(x),x) = 2*x^2+3*y(x); 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+3 x +\frac {9}{2} x^{2}+\frac {9}{2} x^{3}+\frac {27}{8} x^{4}+\frac {81}{40} x^{5}\right ) y \left (0\right )+\frac {2 x^{3}}{3}+\frac {x^{4}}{2}+\frac {3 x^{5}}{10}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 61

ode=D[y[x],x]==2*x^2+3*y[x]; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {3 x^5}{10}+\frac {x^4}{2}+\frac {2 x^3}{3}+c_1 \left (\frac {81 x^5}{40}+\frac {27 x^4}{8}+\frac {9 x^3}{2}+\frac {9 x^2}{2}+3 x+1\right ) \]

✓ Sympy. Time used: 0.846 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**2 - 3*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {x^{3} \left (27 C_{1} + 4\right )}{6} + \frac {x^{4} \left (27 C_{1} + 4\right )}{8} + \frac {3 x^{5} \left (27 C_{1} + 4\right )}{40} + C_{1} + 3 C_{1} x + \frac {9 C_{1} x^{2}}{2} + O\left (x^{6}\right ) \]