32.6.1 problem Exercise 12.1, page 103

Internal problem ID [5866]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.1, page 103
Date solved : Monday, January 27, 2025 at 01:21:40 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 59

dsolve(2*x*y(x)*diff(y(x),x)+(1+x)*y(x)^2=exp(x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {x \,{\mathrm e}^{x} \left ({\mathrm e}^{2 x}+2 c_{1} \right )}\, {\mathrm e}^{-x}}{2 x} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {x \,{\mathrm e}^{x} \left ({\mathrm e}^{2 x}+2 c_{1} \right )}\, {\mathrm e}^{-x}}{2 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.128 (sec). Leaf size: 70

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

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