60.5.39 problem 1576

Internal problem ID [11536]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1576
Date solved : Sunday, March 30, 2025 at 08:24:24 PM
CAS classification : [[_high_order, _missing_x]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 67

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

\[ y = c_1 \,{\mathrm e}^{a x}+c_2 \,{\mathrm e}^{-a x}+c_3 \,{\mathrm e}^{\frac {\left (-\operatorname {df} +\sqrt {a^{2} f^{2}+\operatorname {df}^{2}}\right ) x}{f}}+c_4 \,{\mathrm e}^{-\frac {\left (\operatorname {df} +\sqrt {a^{2} f^{2}+\operatorname {df}^{2}}\right ) x}{f}} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 80

ode=f*(D[y[x],{x,4}]-2*a^2*D[y[x],{x,2}]+a^4*y[x])+2*df*(D[y[x],{x,3}]-a^2*D[y[x],x])==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 e^{\frac {x \left (\sqrt {a^2 f^2+\text {df}^2}-\text {df}\right )}{f}}+c_2 e^{-\frac {x \left (\sqrt {a^2 f^2+\text {df}^2}+\text {df}\right )}{f}}+c_3 e^{-a x}+c_4 e^{a x} \]

✓ Sympy. Time used: 0.429 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
a = symbols("a") 
df = symbols("df") 
f = symbols("f") 
y = Function("y") 
ode = Eq(2*df*(-a**2*Derivative(y(x), x) + Derivative(y(x), (x, 3))) + f*(a**4*y(x) - 2*a**2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- a x} + C_{2} e^{a x} + C_{3} e^{\frac {x \left (- df + \sqrt {a^{2} f^{2} + df^{2}}\right )}{f}} + C_{4} e^{- \frac {x \left (df + \sqrt {a^{2} f^{2} + df^{2}}\right )}{f}} \]