60.3.173 problem 1177

Internal problem ID [11183]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1177
Date solved : Monday, January 27, 2025 at 10:48:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 30

dsolve(x^2*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+(x^2+2)*y(x)-x^2/cos(x)=0,y(x), singsol=all)
 

\[ y = x \left (-\left (\int \frac {\tan \left (x \right )}{x}d x \right ) \cos \left (x \right )+\cos \left (x \right ) c_{1} +\sin \left (x \right ) \left (c_{2} +\ln \left (x \right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 1.025 (sec). Leaf size: 208

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

\[ y(x)\to \frac {\exp \left (\frac {1}{2} \left (-\int _1^x\frac {(K[1]-2) K[1]+2}{K[1]^2}dK[1]+x-\frac {2}{x}\right )\right ) \left (\int _1^x-\frac {\exp \left (\frac {1}{2} \left (K[3]+\int _1^{K[3]}\frac {(K[1]-2) K[1]+2}{K[1]^2}dK[1]-\frac {2}{K[3]}\right )\right ) \sec (K[3]) \int _1^{K[3]}e^{\frac {2}{K[2]}-K[2]} K[2]^2dK[2]}{K[3]}dK[3]+\int _1^xe^{\frac {2}{K[2]}-K[2]} K[2]^2dK[2] \left (\int _1^x\frac {\exp \left (\frac {1}{2} \left (K[4]+\int _1^{K[4]}\frac {(K[1]-2) K[1]+2}{K[1]^2}dK[1]-\frac {2}{K[4]}\right )\right ) \sec (K[4])}{K[4]}dK[4]+c_2\right )+c_1\right )}{x} \]