12.6.14 problem 14

Internal problem ID [1693]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Exact equations. Section 2.5 Page 79
Problem number : 14
Date solved : Monday, January 27, 2025 at 05:29:33 AM
CAS classification : [_exact, [_Abel, `2nd type`, `class B`]]

\begin{align*} {\mathrm e}^{x} \left (x^{2} y^{2}+2 x y^{2}\right )+6 x +\left (2 x^{2} y \,{\mathrm e}^{x}+2\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 67

dsolve((exp(x)*(x^2*y(x)^2+2*x*y(x)^2)+6*x)+(2*x^2*y(x)*exp(x)+2)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {\left (-1+\sqrt {-3 \,{\mathrm e}^{x} x^{4}-{\mathrm e}^{x} c_1 \,x^{2}+1}\right ) {\mathrm e}^{-x}}{x^{2}} \\ y &= \frac {\left (-1-\sqrt {-3 \,{\mathrm e}^{x} x^{4}-{\mathrm e}^{x} c_1 \,x^{2}+1}\right ) {\mathrm e}^{-x}}{x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 35.727 (sec). Leaf size: 76

DSolve[(Exp[x]*(x^2*y[x]^2+2*x*y[x]^2)+6*x)+(2*x^2*y[x]*Exp[x]+2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {e^{-x} \left (1+\sqrt {1+e^x \left (-3 x^4+c_1 x^2\right )}\right )}{x^2} \\ y(x)\to \frac {e^{-x} \left (-1+\sqrt {1+e^x \left (-3 x^4+c_1 x^2\right )}\right )}{x^2} \\ \end{align*}