77.1.160 problem 187 (page 297)
Internal
problem
ID
[17979]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
187
(page
297)
Date
solved
:
Monday, March 31, 2025 at 04:53:03 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d x^{2}}y \left (x \right )+\frac {d}{d x}z \left (x \right )-2 z \left (x \right )&={\mathrm e}^{2 x}\\ \frac {d}{d x}z \left (x \right )+2 \frac {d}{d x}y \left (x \right )-3 y \left (x \right )&=0 \end{align*}
✓ Maple. Time used: 0.227 (sec). Leaf size: 117
ode:=[diff(diff(y(x),x),x)+diff(z(x),x)-2*z(x) = exp(2*x), diff(z(x),x)+2*diff(y(x),x)-3*y(x) = 0];
dsolve(ode);
\begin{align*}
y \left (x \right ) &= \frac {{\mathrm e}^{2 x}}{4}+c_1 \,{\mathrm e}^{x}+c_2 \,{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )+c_3 \,{\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right ) \\
z \left (x \right ) &= -\frac {{\mathrm e}^{2 x}}{8}+c_1 \,{\mathrm e}^{x}-\frac {7 c_2 \,{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{4}+\frac {c_2 \,{\mathrm e}^{\frac {x}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{4}-\frac {7 c_3 \,{\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{4}-\frac {c_3 \,{\mathrm e}^{\frac {x}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{4} \\
\end{align*}
✓ Mathematica. Time used: 2.152 (sec). Leaf size: 199
ode={D[y[x],{x,2}]+D[z[x],x]-2*z[x]==Exp[2*x],D[z[x],x]+2*D[y[x],x]-3*y[x]==0};
ic={};
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{276} e^{x/2} \left (23 e^{x/2} \left (3 e^x+6 c_1+2 c_2+4 c_3\right )+46 (3 c_1-c_2-2 c_3) \cos \left (\frac {\sqrt {23} x}{2}\right )-2 \sqrt {23} (9 c_1-11 c_2+2 c_3) \sin \left (\frac {\sqrt {23} x}{2}\right )\right ) \\
z(x)\to -\frac {1}{552} e^{x/2} \left (23 e^{x/2} \left (3 e^x-4 (3 c_1+c_2+2 c_3)\right )+92 (3 c_1+c_2-4 c_3) \cos \left (\frac {\sqrt {23} x}{2}\right )-4 \sqrt {23} (33 c_1-25 c_2-8 c_3) \sin \left (\frac {\sqrt {23} x}{2}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.538 (sec). Leaf size: 197
from sympy import *
x = symbols("x")
y = Function("y")
z = Function("z")
ode=[Eq(-2*z(x) - exp(2*x) + Derivative(y(x), (x, 2)) + Derivative(z(x), x),0),Eq(-3*y(x) + 2*Derivative(y(x), x) + Derivative(z(x), x),0)]
ics = {}
dsolve(ode,func=[y(x),z(x)],ics=ics)
\[
\left [ y{\left (x \right )} = C_{3} e^{x} + \left (\frac {7 C_{1}}{18} - \frac {\sqrt {23} C_{2}}{18}\right ) e^{\frac {x}{2}} \sin {\left (\frac {\sqrt {23} x}{2} \right )} - \left (\frac {\sqrt {23} C_{1}}{18} + \frac {7 C_{2}}{18}\right ) e^{\frac {x}{2}} \cos {\left (\frac {\sqrt {23} x}{2} \right )} + \frac {e^{2 x} \sin ^{2}{\left (\frac {\sqrt {23} x}{2} \right )}}{12} + \frac {e^{2 x} \cos ^{2}{\left (\frac {\sqrt {23} x}{2} \right )}}{12} + \frac {e^{2 x}}{6}, \ z{\left (x \right )} = - C_{1} e^{\frac {x}{2}} \sin {\left (\frac {\sqrt {23} x}{2} \right )} + C_{2} e^{\frac {x}{2}} \cos {\left (\frac {\sqrt {23} x}{2} \right )} + C_{3} e^{x} - \frac {7 e^{2 x} \sin ^{2}{\left (\frac {\sqrt {23} x}{2} \right )}}{24} - \frac {7 e^{2 x} \cos ^{2}{\left (\frac {\sqrt {23} x}{2} \right )}}{24} + \frac {e^{2 x}}{6}\right ]
\]