75.7.8 problem 182

Internal problem ID [16801]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 182
Date solved : Tuesday, January 28, 2025 at 09:29:30 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 64

dsolve(( x*y(x)/sqrt(1+x^2) + 2*x*y(x) -y(x)/x  )+(  sqrt(1+x^2) + x^2-ln(x) )*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y = c_{1} {\mathrm e}^{-\int \frac {2 \sqrt {x^{2}+1}\, x^{2}+x^{2}-\sqrt {x^{2}+1}}{\sqrt {x^{2}+1}\, x \left (\sqrt {x^{2}+1}+x^{2}-\ln \left (x \right )\right )}d x} \]

✓ Solution by Mathematica

Time used: 1.719 (sec). Leaf size: 94

DSolve[( x*y[x]/Sqrt[1+x^2] + 2*x*y[x] -y[x]/x  )+(  Sqrt[1+x^2] + x^2-Log[x] )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 \exp \left (\int _1^x\frac {\sqrt {K[1]^2+1}-K[1]^2 \left (2 \sqrt {K[1]^2+1}+1\right )}{K[1] \left (\left (\sqrt {K[1]^2+1}+1\right ) K[1]^2-\sqrt {K[1]^2+1} \log (K[1])+1\right )}dK[1]\right ) \\ y(x)\to 0 \\ \end{align*}