61.34.6 problem 6
Internal
problem
ID
[12691]
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
:
6
Date
solved
:
Monday, March 31, 2025 at 06:51:24 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+c \,{\mathrm e}^{2 \lambda x}-\frac {\lambda ^{2}}{4}\right ) y&=0 \end{align*}
✓ Maple. Time used: 0.199 (sec). Leaf size: 193
ode:=diff(diff(y(x),x),x)+(a*exp(4*lambda*x)+b*exp(3*lambda*x)+c*exp(2*lambda*x)-1/4*lambda^2)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (2 a c_2 \operatorname {hypergeom}\left (\left [\frac {12 \lambda \,a^{{3}/{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \,{\mathrm e}^{\lambda x}+b \right )^{2}}{4 \lambda \,a^{{3}/{2}}}\right ) {\mathrm e}^{\lambda x}+\operatorname {hypergeom}\left (\left [\frac {12 \lambda \,a^{{3}/{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \,{\mathrm e}^{\lambda x}+b \right )^{2}}{4 \lambda \,a^{{3}/{2}}}\right ) c_2 b +c_1 \operatorname {hypergeom}\left (\left [\frac {4 \lambda \,a^{{3}/{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{{3}/{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \,{\mathrm e}^{\lambda x}+b \right )^{2}}{4 \lambda \,a^{{3}/{2}}}\right )\right ) {\mathrm e}^{-\frac {\lambda ^{2} x \sqrt {a}+i {\mathrm e}^{2 \lambda x} a +i {\mathrm e}^{\lambda x} b}{2 \lambda \sqrt {a}}}
\]
✓ Mathematica. Time used: 0.973 (sec). Leaf size: 189
ode=D[y[x],{x,2}]+(a*Exp[4*\[Lambda]*x]+b*Exp[3*\[Lambda]*x]+c*Exp[2*\[Lambda]*x]-1/4*\[Lambda]^2)*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {e^{-\frac {\frac {i b e^{\lambda x}}{\sqrt {a}}+i \sqrt {a} e^{2 \lambda x}+\lambda }{2 \lambda }} \left (c_1 \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c+4 i a^{3/2} \lambda \right )}{8 a^{3/2} \lambda },\frac {\sqrt [4]{-1} \left (2 e^{x \lambda } a+b\right )}{2 a^{3/4} \sqrt {\lambda }}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+4 a^{3/2} \lambda }{16 a^{3/2} \lambda },\frac {1}{2},\frac {i \left (2 e^{x \lambda } a+b\right )^2}{4 a^{3/2} \lambda }\right )\right )}{\sqrt {e^{\lambda x}}}
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq((a*exp(4*lambda_*x) + b*exp(3*lambda_*x) + c*exp(2*lambda_*x) - lambda_**2/4)*y(x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False