12.6.24 problem 24

Internal problem ID [1703]
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 : 24
Date solved : Monday, January 27, 2025 at 05:31:02 AM
CAS classification : [_exact, _Bernoulli]

\begin{align*} {\mathrm e}^{x} \left (x^{4} y^{2}+4 x^{3} y^{2}+1\right )+\left (2 x^{4} y \,{\mathrm e}^{x}+2 y\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 64

dsolve((exp(x)*(x^4*y(x)^2+4*x^3*y(x)^2+1))+(2*x^4*y(x)*exp(x)+2*y(x))*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.146 (sec). Leaf size: 64

DSolve[(Exp[x]*(x^4*y[x]^2+4*x^3*y[x]^2+1))+(2*x^4*y[x]*Exp[x]+2*y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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