50.3.12 problem 12
Internal
problem
ID
[10156]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
12
Date
solved
:
Tuesday, September 30, 2025 at 07:05:21 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+y^{\prime }+y&=\sin \left (x \right ) \end{align*}
With initial conditions
\begin{align*}
y^{\prime }\left (1\right )&=0 \\
\end{align*}
✓ Maple. Time used: 0.187 (sec). Leaf size: 110
ode:=diff(diff(y(x),x),x)+diff(y(x),x)+y(x) = sin(x);
ic:=[D(y)(1) = 0];
dsolve([ode,op(ic)],y(x), singsol=all);
\[
y = \frac {2 \sin \left (1\right ) {\mathrm e}^{\frac {1}{2}-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{-\frac {x}{2}} c_2 \left (\cos \left (\frac {\sqrt {3}}{2}\right ) \sqrt {3}-\sin \left (\frac {\sqrt {3}}{2}\right )\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\left (\sin \left (\frac {\sqrt {3}}{2}\right ) \sqrt {3}+\cos \left (\frac {\sqrt {3}}{2}\right )\right ) \left ({\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_2 -\cos \left (x \right )\right )}{\sin \left (\frac {\sqrt {3}}{2}\right ) \sqrt {3}+\cos \left (\frac {\sqrt {3}}{2}\right )}
\]
✓ Mathematica. Time used: 0.206 (sec). Leaf size: 578
ode=D[y[x],{x,3}]+D[y[x],x]+y[x]==Sin[x];
ic={Derivative[1][y][1] == 0};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Sympy. Time used: 0.144 (sec). Leaf size: 133
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x) - sin(x) + Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {Subs(Derivative(y(x), x), x, 1): 0}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (C_{2} \cos {\left (\frac {\sqrt {3} x}{2} \right )} + \left (C_{2} \left (\frac {\cos {\left (\frac {\sqrt {3}}{2} \right )}}{- \sin {\left (\frac {\sqrt {3}}{2} \right )} + \sqrt {3} \cos {\left (\frac {\sqrt {3}}{2} \right )}} + \frac {\sqrt {3} \sin {\left (\frac {\sqrt {3}}{2} \right )}}{- \sin {\left (\frac {\sqrt {3}}{2} \right )} + \sqrt {3} \cos {\left (\frac {\sqrt {3}}{2} \right )}}\right ) - \frac {2 e^{\frac {1}{2}} \sin {\left (1 \right )}}{- \sin {\left (\frac {\sqrt {3}}{2} \right )} + \sqrt {3} \cos {\left (\frac {\sqrt {3}}{2} \right )}}\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} - \cos {\left (x \right )}
\]