60.3.80 problem 1084

Internal problem ID [11090]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1084
Date solved : Tuesday, January 28, 2025 at 05:40:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.194 (sec). Leaf size: 20

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

\[ y = f \left (\operatorname {BesselJ}\left (v , g \left (x \right )\right ) c_{1} +\operatorname {BesselY}\left (v , g \left (x \right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.140 (sec). Leaf size: 35

DSolve[-(D[y[x],x]*((2*Derivative[1][f][x])/f[x] - Derivative[1][g][x]/g[x] + Derivative[2][g][x]/Derivative[1][g][x])) + y[x]*(Derivative[1][g][x]^2 - (v^2*Derivative[1][g][x]^2)/g[x]^2 - Derivative[2][f][x]/f[x] + (Derivative[1][f][x]*((2*Derivative[1][f][x])/f[x] - Derivative[1][g][x]/g[x] + Derivative[2][g][x]/Derivative[1][g][x]))/f[x]) + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to f(x) \left (c_1 \operatorname {BesselJ}\left (\sqrt {v^2},g(x)\right )+c_2 \operatorname {BesselY}\left (\sqrt {v^2},g(x)\right )\right ) \]