2.35.21 Problem 21
Internal
problem
ID
[13945]
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
:
21
Date
solved
:
Friday, December 19, 2025 at 08:51:54 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*}
y^{\prime \prime }+\left (a +b \,{\mathrm e}^{2 \lambda x}\right ) y^{\prime }+\lambda \left (a -\lambda -b \,{\mathrm e}^{2 \lambda x}\right ) y&=0 \\
\end{align*}
2.35.21.1 ✗ Maple
ode:=diff(diff(y(x),x),x)+(a+b*exp(2*lambda*x))*diff(y(x),x)+lambda*(a-lambda-b*exp(2*lambda*x))*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[ \text {No solution found} \]
2.35.21.2 ✓ Mathematica. Time used: 0.123 (sec). Leaf size: 248
ode=D[y[x],{x,2}]+(a+b*Exp[2*\[Lambda]*x])*D[y[x],x]+\[Lambda]*(a-\[Lambda]-b*Exp[2*\[Lambda]*x])*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {-\frac {1}{2} c_2 (a-2 \lambda ) e^{-\frac {b e^{2 \lambda x}}{2 \lambda }} \left (b \lambda e^{2 \lambda x}\right )^{-\frac {a}{2 \lambda }} \left (b 2^{\frac {a}{2 \lambda }} \lambda ^{a/\lambda } e^{2 \lambda x}+\operatorname {Gamma}\left (1-\frac {a}{2 \lambda }\right ) e^{\frac {b e^{2 \lambda x}}{2 \lambda }} \left (a+b e^{2 \lambda x}\right ) \left (b \lambda e^{2 \lambda x}\right )^{\frac {a}{2 \lambda }}-e^{\frac {b e^{2 \lambda x}}{2 \lambda }} \left (a+b e^{2 \lambda x}\right ) \left (b \lambda e^{2 \lambda x}\right )^{\frac {a}{2 \lambda }} \Gamma \left (1-\frac {a}{2 \lambda },\frac {b e^{2 x \lambda }}{2 \lambda }\right )\right )-\frac {2 i c_1 \lambda ^2 \left (a+b e^{2 \lambda x}\right )}{a}}{\sqrt {2} \lambda \sqrt {b \lambda e^{2 \lambda x}}} \end{align*}
2.35.21.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(lambda_*(a - b*exp(2*lambda_*x) - lambda_)*y(x) + (a + b*exp(2*lambda_*x))*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False