88.11.7 problem 8

Internal problem ID [24054]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 2. Differential equations of first order. Exercise at page 54
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:55:16 PM
CAS classification : system_of_ODEs

\begin{align*} y^{\prime }&=x^{2}+6 y+4 z \left (x \right )\\ z^{\prime }\left (x \right )&=y+3 z \left (x \right ) \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 53

ode:=[diff(y(x),x) = x^2+6*y(x)+4*z(x), diff(z(x),x) = y(x)+3*z(x)]; 
dsolve(ode);
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{2 x} c_2 +{\mathrm e}^{7 x} c_1 -\frac {3 x^{2}}{14}-\frac {13 x}{98}-\frac {75}{1372} \\ z \left (x \right ) &= \frac {x^{2}}{14}-{\mathrm e}^{2 x} c_2 +\frac {{\mathrm e}^{7 x} c_1}{4}+\frac {9 x}{98}+\frac {67}{1372} \\ \end{align*}

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 100

ode={D[y[x],x]==x^2+6*y[x]+4*z[x],D[z[x],x]==y[x]+3*z[x]}; 
ic={}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {3 x^2}{14}-\frac {13 x}{98}+\frac {1}{5} (c_1-4 c_2) e^{2 x}+\frac {4}{5} (c_1+c_2) e^{7 x}-\frac {75}{1372}\\ z(x)&\to \frac {x^2}{14}+\frac {9 x}{98}-\frac {1}{5} (c_1-4 c_2) e^{2 x}+\frac {1}{5} (c_1+c_2) e^{7 x}+\frac {67}{1372} \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x**2 - 6*y(x) - 4*z(x) + Derivative(y(x), x),0),Eq(-y(x) - 3*z(x) + Derivative(z(x), x),0)] 
ics = {} 
dsolve(ode,func=[y(x),z(x)],ics=ics)
 

\[ \left [ y{\left (x \right )} = - C_{1} e^{2 x} + 4 C_{2} e^{7 x} - \frac {3 x^{2}}{14} - \frac {13 x}{98} - \frac {75}{1372}, \ z{\left (x \right )} = C_{1} e^{2 x} + C_{2} e^{7 x} + \frac {x^{2}}{14} + \frac {9 x}{98} + \frac {67}{1372}\right ] \]