29.8.28 problem 233
Internal
problem
ID
[4833]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
8
Problem
number
:
233
Date
solved
:
Sunday, March 30, 2025 at 04:02:48 AM
CAS
classification
:
[_linear]
\begin{align*} \left (a +x \right ) y^{\prime }&=b x +c y \end{align*}
✓ Maple. Time used: 0.001 (sec). Leaf size: 28
ode:=(x+a)*diff(y(x),x) = b*x+c*y(x);
dsolve(ode,y(x), singsol=all);
\[
y = \left (a +x \right )^{c} c_1 -\frac {b \left (x c +a \right )}{c \left (c -1\right )}
\]
✓ Mathematica. Time used: 0.586 (sec). Leaf size: 32
ode=(a+x) D[y[x],x]==b x+c y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {a b+b c x}{c-c^2}+c_1 (a+x)^c
\]
✓ Sympy. Time used: 12.590 (sec). Leaf size: 400
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(-b*x - c*y(x) + (a + x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} \frac {C_{1} c^{2} e^{c \log {\left (a + x \right )}}}{c^{2} - c} - \frac {C_{1} c e^{c \log {\left (a + x \right )}}}{c^{2} - c} - \frac {a b}{c^{2} - c} - \frac {b c x}{c^{2} - c} & \text {for}\: c > -\infty \wedge c < \infty \wedge c \neq 0 \wedge c \neq 1 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {C_{1} c^{2} e^{c \log {\left (a + x \right )}}}{c^{3} e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} - c^{2} e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} + 2 c^{2} - 2 c} - \frac {C_{1} c e^{c \log {\left (a + x \right )}}}{c^{3} e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} - c^{2} e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} + 2 c^{2} - 2 c} - \frac {a b}{c^{3} e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} - c^{2} e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} + 2 c^{2} - 2 c} - \frac {b c x}{c^{3} e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} - c^{2} e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} + 2 c^{2} - 2 c} & \text {for}\: \left (c \geq \infty \vee c \leq -\infty \right ) \wedge c \neq 0 \wedge c \neq 1 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {C_{1} e^{c \log {\left (a + x \right )}}}{c e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} + 1} - \frac {a b e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )}}{c e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} + 1} + \frac {b x e^{c \log {\left (a + x \right )}}}{c e^{c \log {\left (a + x \right )}} \log {\left (a + x \right )} + 1} & \text {for}\: c = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} a + C_{1} x + a b \log {\left (a + x \right )} + a b + b x \log {\left (a + x \right )} & \text {for}\: c = 1 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]