Internal problem ID [3152]
Internal file name [OUTPUT/2644_Sunday_June_05_2022_08_38_01_AM_44925495/index.tex]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 7.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {x y^{3}+\left (y+1\right ) {\mathrm e}^{-x} y^{\prime }=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {x \,y^{3} {\mathrm e}^{x}}{y +1} \end {align*}
Where \(f(x)=-{\mathrm e}^{x} x\) and \(g(y)=\frac {y^{3}}{y +1}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {y^{3}}{y +1}} \,dy &= -{\mathrm e}^{x} x \,d x \\ \int { \frac {1}{\frac {y^{3}}{y +1}} \,dy} &= \int {-{\mathrm e}^{x} x \,d x} \\ \frac {-y -\frac {1}{2}}{y^{2}}&=-\left (-1+x \right ) {\mathrm e}^{x}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= -\frac {{\mathrm e}^{-x}-\sqrt {-2 \,{\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{-2 x}+2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}}}{2 \left (c_{1} {\mathrm e}^{-x}-x +1\right )} \\ y &= -\frac {{\mathrm e}^{-x}+\sqrt {-2 \,{\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{-2 x}+2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}}}{2 \left (c_{1} {\mathrm e}^{-x}-x +1\right )} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {{\mathrm e}^{-x}-\sqrt {-2 \,{\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{-2 x}+2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}}}{2 \left (c_{1} {\mathrm e}^{-x}-x +1\right )} \\ \tag{2} y &= -\frac {{\mathrm e}^{-x}+\sqrt {-2 \,{\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{-2 x}+2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}}}{2 \left (c_{1} {\mathrm e}^{-x}-x +1\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {{\mathrm e}^{-x}-\sqrt {-2 \,{\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{-2 x}+2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}}}{2 \left (c_{1} {\mathrm e}^{-x}-x +1\right )} \] Verified OK.
\[ y = -\frac {{\mathrm e}^{-x}+\sqrt {-2 \,{\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{-2 x}+2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}}}{2 \left (c_{1} {\mathrm e}^{-x}-x +1\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{3}+\left (y+1\right ) {\mathrm e}^{-x} y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x y^{3}}{\left (y+1\right ) {\mathrm e}^{-x}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } \left (y+1\right )}{y^{3}}=-\frac {x}{{\mathrm e}^{-x}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } \left (y+1\right )}{y^{3}}d x =\int -\frac {x}{{\mathrm e}^{-x}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}-\frac {1}{2 y^{2}}=-\frac {-1+x}{{\mathrm e}^{-x}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-\frac {{\mathrm e}^{-x}-\sqrt {-2 \left ({\mathrm e}^{-x}\right )^{2} c_{1} +\left ({\mathrm e}^{-x}\right )^{2}+2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}}}{2 \left (c_{1} {\mathrm e}^{-x}-x +1\right )}, y=-\frac {{\mathrm e}^{-x}+\sqrt {-2 \left ({\mathrm e}^{-x}\right )^{2} c_{1} +\left ({\mathrm e}^{-x}\right )^{2}+2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}}}{2 \left (c_{1} {\mathrm e}^{-x}-x +1\right )}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 73
dsolve(x*y(x)^3+(y(x)+1)*exp(-x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1-\sqrt {\left (2 x -2\right ) {\mathrm e}^{x}+2 c_{1} +1}}{\left (2 x -2\right ) {\mathrm e}^{x}+2 c_{1}} \\ y \left (x \right ) &= \frac {1+\sqrt {\left (2 x -2\right ) {\mathrm e}^{x}+2 c_{1} +1}}{\left (2 x -2\right ) {\mathrm e}^{x}+2 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 9.963 (sec). Leaf size: 88
DSolve[x*y[x]^3+(y[x]+1)*Exp[-x]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1-\sqrt {2 e^x (x-1)+1-2 c_1}}{2 e^x (x-1)-2 c_1} \\ y(x)\to \frac {1+\sqrt {2 e^x (x-1)+1-2 c_1}}{2 e^x (x-1)-2 c_1} \\ y(x)\to 0 \\ \end{align*}