56.1.80 problem 79

Internal problem ID [8792]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 79
Date solved : Monday, January 27, 2025 at 04:58:48 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 36

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

\[ y = c_{1} \left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a^{2} \left (x^{2}+1\right )}\right )^{\frac {1}{\sqrt {a^{2}}}} \sqrt {x^{2}+1} \]

✓ Solution by Mathematica

Time used: 0.298 (sec). Leaf size: 116

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

\begin{align*} y(x)\to c_1 \left (a \left (-\sqrt {a^2 \left (x^2+1\right )}+\sqrt {a^2}+a x\right )\right )^{-\frac {a+1}{a}} \left (a \left (\sqrt {a^2 \left (x^2+1\right )}-\sqrt {a^2}+a x\right )\right )^{\frac {1}{a}-1} \left (\sqrt {a^2} \sqrt {a^2 \left (x^2+1\right )}-a^2 \left (x^2+1\right )\right ) \\ y(x)\to 0 \\ \end{align*}