60.4.68 problem 1518

Internal problem ID [11519]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1518
Date solved : Tuesday, January 28, 2025 at 06:06:43 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) x y^{\prime \prime \prime }+3 \left (2 x^{2}+1\right ) y^{\prime \prime }-12 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.025 (sec). Leaf size: 60

dsolve((x^2+1)*x*diff(diff(diff(y(x),x),x),x)+3*(2*x^2+1)*diff(diff(y(x),x),x)-12*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {3 \sqrt {x^{2}+1}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right ) c_{2} x^{2}+c_{1} x^{2} \sqrt {x^{2}+1}+2 c_3 \,x^{3}-3 c_{2} x^{2}+x c_3 -c_{2}}{x} \]

✓ Solution by Mathematica

Time used: 0.204 (sec). Leaf size: 270

DSolve[-12*y[x] + 3*(1 + 2*x^2)*D[y[x],{x,2}] + x*(1 + x^2)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{3} \left (2 x^2+1\right ) \left (c_2 \int _1^x\exp \left (\int _1^{K[4]}\frac {4 K[1]^4+6 K[1]^2+3}{4 K[1]^5+6 K[1]^3+2 K[1]}dK[1]-\frac {1}{2} \int _1^{K[4]}\frac {24 \left (K[2]^4+K[2]^2\right )+3}{2 K[2]^5+3 K[2]^3+K[2]}dK[2]\right )dK[4]+c_3 \int _1^x\exp \left (\int _1^{K[5]}\frac {4 K[1]^4+6 K[1]^2+3}{4 K[1]^5+6 K[1]^3+2 K[1]}dK[1]-\frac {1}{2} \int _1^{K[5]}\frac {24 \left (K[2]^4+K[2]^2\right )+3}{2 K[2]^5+3 K[2]^3+K[2]}dK[2]\right ) \int _1^{K[5]}\exp \left (-2 \int _1^{K[3]}\frac {4 K[1]^4+6 K[1]^2+3}{4 K[1]^5+6 K[1]^3+2 K[1]}dK[1]\right )dK[3]dK[5]+c_1\right ) \]