56.1.86 problem 85

Internal problem ID [8798]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 85
Date solved : Monday, January 27, 2025 at 05:01:35 PM
CAS classification : [_Bernoulli]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 9

dsolve([diff(y(x),x)= 2*y(x)*(x*sqrt(y(x)) - 1),y(0) = 1],y(x), singsol=all)
 

\[ y = \frac {1}{\left (x +1\right )^{2}} \]

✓ Solution by Mathematica

Time used: 0.682 (sec). Leaf size: 20

DSolve[{D[y[x],x]==2*y[x]*(x*Sqrt[y[x]-1]),{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 1 \\ y(x)\to \sec ^2\left (\frac {x^2}{2}\right ) \\ \end{align*}