62.36.8 problem Ex 8

Internal problem ID [13006]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 60. Exact equation. Integrating factor. Page 139
Problem number : Ex 8
Date solved : Tuesday, January 28, 2025 at 08:24:46 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 x^{3} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 22

dsolve(x^2*diff(y(x),x$3)-5*x*diff(y(x),x$2)+(4*x^4+5)*diff(y(x),x)-8*x^3*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} x^{2}+c_{2} \cos \left (x^{2}\right )+c_{3} \sin \left (x^{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.493 (sec). Leaf size: 98

DSolve[x^2*D[y[x],{x,3}]-5*x*D[y[x],{x,2}]+(4*x^4+5)*D[y[x],x]-8*x^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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