60.5.41 problem 1576

Internal problem ID [11575]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1576
Date solved : Monday, January 27, 2025 at 11:23:35 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*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 67

dsolve(f*(diff(y(x),x$4)-2*a^2*diff(y(x),x$2)+a^4*y(x))+2*df*(diff(y(x),x$3)-a^2*diff(y(x),x))=0,y(x), singsol=all)
 

\[ y = c_{1} {\mathrm e}^{a x}+c_{2} {\mathrm e}^{-a x}+c_3 \,{\mathrm e}^{\frac {\left (-\operatorname {df} +\sqrt {a^{2} f^{2}+\operatorname {df}^{2}}\right ) x}{f}}+c_4 \,{\mathrm e}^{-\frac {\left (\operatorname {df} +\sqrt {a^{2} f^{2}+\operatorname {df}^{2}}\right ) x}{f}} \]

✓ Solution by Mathematica

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

DSolve[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,y[x],x,IncludeSingularSolutions -> True]
 

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