60.3.60 problem 1061

Internal problem ID [11070]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1061
Date solved : Monday, January 27, 2025 at 10:43:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime } \sqrt {x}+\left (\frac {1}{4 \sqrt {x}}+\frac {x}{4}-9\right ) y-x \,{\mathrm e}^{-\frac {x^{{3}/{2}}}{3}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 28

dsolve(diff(diff(y(x),x),x)+diff(y(x),x)*x^(1/2)+(1/4/x^(1/2)+1/4*x-9)*y(x)-x*exp(-1/3*x^(3/2))=0,y(x), singsol=all)
 

\[ y = -\frac {{\mathrm e}^{-\frac {x^{{3}/{2}}}{3}} \left (-9 \cosh \left (3 x \right ) c_{1} -9 \sinh \left (3 x \right ) c_{2} +x \right )}{9} \]

✓ Solution by Mathematica

Time used: 0.135 (sec). Leaf size: 79

DSolve[-(x/E^(x^(3/2)/3)) + (-9 + 1/(4*Sqrt[x]) + x/4)*y[x] + Sqrt[x]*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{6} e^{-\frac {1}{3} \left (\sqrt {x}+9\right ) x} \left (6 \int _1^x-\frac {1}{6} e^{3 K[1]} K[1]dK[1]+e^{6 x} \int _1^xe^{-3 K[2]} K[2]dK[2]+c_2 e^{6 x}+6 c_1\right ) \]