60.5.4 problem 1537

Internal problem ID [11538]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1537
Date solved : Monday, January 27, 2025 at 11:23:17 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-12 y^{\prime \prime }+12 y-16 x^{4} {\mathrm e}^{x^{2}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 67

dsolve(diff(diff(diff(diff(y(x),x),x),x),x)-12*diff(diff(y(x),x),x)+12*y(x)-16*x^4*exp(x^2)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{x^{2}}+c_{1} {\mathrm e}^{\sqrt {6-2 \sqrt {6}}\, x}+c_{2} {\mathrm e}^{\sqrt {6+2 \sqrt {6}}\, x}+c_3 \,{\mathrm e}^{-\sqrt {6-2 \sqrt {6}}\, x}+c_4 \,{\mathrm e}^{-\sqrt {6+2 \sqrt {6}}\, x} \]

✓ Solution by Mathematica

Time used: 1.037 (sec). Leaf size: 359

DSolve[-16*E^x^2*x^4 + 12*y[x] - 12*D[y[x],{x,2}] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{\sqrt {6-2 \sqrt {6}} x} \int _1^x-\frac {1}{3} \sqrt {3+\sqrt {6}} e^{-\left (\left (\sqrt {6-2 \sqrt {6}}-K[1]\right ) K[1]\right )} K[1]^4dK[1]+e^{-\sqrt {6-2 \sqrt {6}} x} \int _1^x\frac {1}{3} \sqrt {3+\sqrt {6}} e^{K[2] \left (K[2]+\sqrt {6-2 \sqrt {6}}\right )} K[2]^4dK[2]+e^{\sqrt {2 \left (3+\sqrt {6}\right )} x} \int _1^x\frac {1}{3} \sqrt {3-\sqrt {6}} e^{K[3] \left (K[3]-\sqrt {2 \left (3+\sqrt {6}\right )}\right )} K[3]^4dK[3]+e^{-\sqrt {2 \left (3+\sqrt {6}\right )} x} \int _1^x-\frac {1}{3} \sqrt {3-\sqrt {6}} e^{K[4] \left (K[4]+\sqrt {2 \left (3+\sqrt {6}\right )}\right )} K[4]^4dK[4]+c_1 e^{\sqrt {6-2 \sqrt {6}} x}+c_2 e^{-\sqrt {6-2 \sqrt {6}} x}+c_3 e^{\sqrt {2 \left (3+\sqrt {6}\right )} x}+c_4 e^{-\sqrt {2 \left (3+\sqrt {6}\right )} x} \]