60.3.31 problem 1036
Internal
problem
ID
[11027]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1036
Date
solved
:
Sunday, March 30, 2025 at 07:39:37 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+a y^{\prime }+b y-f \left (x \right )&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 110
ode:=diff(diff(y(x),x),x)+a*diff(y(x),x)+b*y(x)-f(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {{\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}} \left (\left (\sqrt {a^{2}-4 b}\, c_2 +\int f \left (x \right ) {\mathrm e}^{-\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right ) {\mathrm e}^{x \sqrt {a^{2}-4 b}}+c_1 \sqrt {a^{2}-4 b}-\int f \left (x \right ) {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right )}{\sqrt {a^{2}-4 b}}
\]
✓ Mathematica. Time used: 0.181 (sec). Leaf size: 152
ode=-f[x] + b*y[x] + a*D[y[x],x] + D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{-\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \left (\int _1^x-\frac {e^{\frac {1}{2} \left (a+\sqrt {a^2-4 b}\right ) K[1]} f(K[1])}{\sqrt {a^2-4 b}}dK[1]+e^{x \sqrt {a^2-4 b}} \int _1^x\frac {e^{\frac {1}{2} \left (a-\sqrt {a^2-4 b}\right ) K[2]} f(K[2])}{\sqrt {a^2-4 b}}dK[2]+c_2 e^{x \sqrt {a^2-4 b}}+c_1\right )
\]
✓ Sympy. Time used: 3.658 (sec). Leaf size: 148
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
f = Function("f")
ode = Eq(a*Derivative(y(x), x) + b*y(x) - f(x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{\frac {x \left (- a + \sqrt {a^{2} - 4 b}\right )}{2}} + C_{2} e^{- \frac {x \left (a + \sqrt {a^{2} - 4 b}\right )}{2}} + \frac {e^{\frac {x \left (- a + \sqrt {a^{2} - 4 b}\right )}{2}} \int f{\left (x \right )} e^{\frac {a x}{2}} e^{- \frac {x \sqrt {a^{2} - 4 b}}{2}}\, dx}{\sqrt {a^{2} - 4 b}} - \frac {e^{- \frac {x \left (a + \sqrt {a^{2} - 4 b}\right )}{2}} \int f{\left (x \right )} e^{\frac {a x}{2}} e^{\frac {x \sqrt {a^{2} - 4 b}}{2}}\, dx}{\sqrt {a^{2} - 4 b}}
\]