23.3.99 problem 101

Internal problem ID [5813]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 101
Date solved : Tuesday, September 30, 2025 at 02:03:38 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \left (a^{2}+b^{2}\right )^{2} y-4 a b y^{\prime }+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

ode:=(a^2+b^2)^2*y(x)-4*a*b*diff(y(x),x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{2 a b x} \left (-c_1 \sin \left (\left (a^{2}-b^{2}\right ) x \right )+c_2 \cos \left (\left (a^{2}-b^{2}\right ) x \right )\right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 62

ode=(a^2 + b^2)^2*y[x] - 4*a*b*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{2 a b x-x \sqrt {-\left (a^2-b^2\right )^2}} \left (c_2 e^{2 x \sqrt {-\left (a^2-b^2\right )^2}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-4*a*b*Derivative(y(x), x) + (a**2 + b**2)**2*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{x \left (- i a^{2} + 2 a b + i b^{2}\right )} + C_{2} e^{x \left (i a^{2} + 2 a b - i b^{2}\right )} \]