2.35.6 Problem 6

Internal problem ID [13930]
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 : Friday, December 19, 2025 at 08:50:23 PM
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*}

2.35.6.1 ✓ Maple. Time used: 0.042 (sec). Leaf size: 204

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);
 

\[ y = \left (2 a c_2 \operatorname {hypergeom}\left (\left [\frac {12 \lambda \,a^{{3}/{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \,{\mathrm e}^{\lambda x}+b \right )^{2}}{4 \lambda \,a^{{3}/{2}}}\right ) {\mathrm e}^{\lambda x}+\operatorname {hypergeom}\left (\left [\frac {12 \lambda \,a^{{3}/{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \,{\mathrm e}^{\lambda x}+b \right )^{2}}{4 \lambda \,a^{{3}/{2}}}\right ) c_2 b +c_1 \operatorname {hypergeom}\left (\left [\frac {4 \lambda \,a^{{3}/{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{{3}/{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \,{\mathrm e}^{\lambda x}+b \right )^{2}}{4 \lambda \,a^{{3}/{2}}}\right )\right ) {\mathrm e}^{-\frac {\lambda ^{2} x \sqrt {a}+i {\mathrm e}^{\lambda x} b +i {\mathrm e}^{2 \lambda x} a}{2 \lambda \sqrt {a}}} \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ 
@ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x),\ 
 dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
   trying a quadrature 
   checking if the LODE has constant coefficients 
   checking if the LODE is of Euler type 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Trying a Liouvillian solution using Kovacics algorithm 
   <- No Liouvillian solutions exists 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Whittaker 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      -> hypergeometric 
         -> heuristic approach 
         -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
         <- hyper3 successful: indirect Equivalence to 0F1 under ``^ @ Moebius`\ 
` is resolved 
      <- hypergeometric successful 
   <- special function solution successful 
   Change of variables used: 
      [x = ln(t)/lambda] 
   Linear ODE actually solved: 
      (4*a*t^4+4*b*t^3+4*c*t^2-lambda^2)*u(t)+4*lambda^2*t*diff(u(t),t)+4*lambd\ 
a^2*t^2*diff(diff(u(t),t),t) = 0 
<- change of variables successful
 

2.35.6.2 ✓ Mathematica. Time used: 0.529 (sec). Leaf size: 178

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]
 

\begin{align*} y(x)&\to \frac {e^{-\frac {i e^{\lambda x} \left (a e^{\lambda x}+b\right )}{2 \sqrt {a} \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}}} \end{align*}

2.35.6.3 ✗ 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