61.34.22 problem 22
Internal
problem
ID
[12707]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.3-1.
Equations
with
exponential
functions
Problem
number
:
22
Date
solved
:
Friday, March 14, 2025 at 12:16:58 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y&=0 \end{align*}
✓ Maple. Time used: 1.066 (sec). Leaf size: 205
ode:=diff(diff(y(x),x),x)+(a+b*exp(lambda*x)+b-3*lambda)*diff(y(x),x)+a^2*lambda*(b-lambda)*exp(2*lambda*x)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {{\mathrm e}^{\lambda x} \left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right )}{2 \lambda }} \left (\operatorname {KummerU}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (-2 \lambda +b +a \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {-2 \lambda +b +a}{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} +\operatorname {KummerM}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (-2 \lambda +b +a \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {-2 \lambda +b +a}{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{1} \right )
\]
✓ Mathematica. Time used: 2.16 (sec). Leaf size: 260
ode=D[y[x],{x,2}]+(a+b*Exp[\[Lambda]*x]+b-3*\[Lambda])*D[y[x],x]+a^2*\[Lambda]*(b-\[Lambda])*Exp[2*\[Lambda]*x]*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \exp \left (-\frac {e^{\lambda x} \left (\sqrt {-4 a^2 b \lambda +4 a^2 \lambda ^2+b^2}+b\right )}{2 \lambda }\right ) \left (c_1 \operatorname {HypergeometricU}\left (\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}},\frac {a+b-2 \lambda }{\lambda },\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )+c_2 L_{-\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}}^{\frac {a+b-3 \lambda }{\lambda }}\left (\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
cg = symbols("cg")
y = Function("y")
ode = Eq(a**2*cg*(b - cg)*y(x)*exp(2*cg*x) + (a + b*exp(cg*x) + b - 3*cg)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False