Internal problem ID [3150]
Internal file name [OUTPUT/2642_Sunday_June_05_2022_08_37_59_AM_14399028/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: 5.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {x y^{3}+{\mathrm e}^{x^{2}} y^{\prime }=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -x \,y^{3} {\mathrm e}^{-x^{2}} \end {align*}
Where \(f(x)=-x \,{\mathrm e}^{-x^{2}}\) and \(g(y)=y^{3}\). Integrating both sides gives \begin{align*} \frac {1}{y^{3}} \,dy &= -x \,{\mathrm e}^{-x^{2}} \,d x \\ \int { \frac {1}{y^{3}} \,dy} &= \int {-x \,{\mathrm e}^{-x^{2}} \,d x} \\ -\frac {1}{2 y^{2}}&=\frac {{\mathrm e}^{-x^{2}}}{2}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \frac {\sqrt {-\left (2 c_{1} {\mathrm e}^{x^{2}}+1\right ) {\mathrm e}^{x^{2}}}}{2 c_{1} {\mathrm e}^{x^{2}}+1} \\ y &= -\frac {\sqrt {-\left (2 c_{1} {\mathrm e}^{x^{2}}+1\right ) {\mathrm e}^{x^{2}}}}{2 c_{1} {\mathrm e}^{x^{2}}+1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {-\left (2 c_{1} {\mathrm e}^{x^{2}}+1\right ) {\mathrm e}^{x^{2}}}}{2 c_{1} {\mathrm e}^{x^{2}}+1} \\ \tag{2} y &= -\frac {\sqrt {-\left (2 c_{1} {\mathrm e}^{x^{2}}+1\right ) {\mathrm e}^{x^{2}}}}{2 c_{1} {\mathrm e}^{x^{2}}+1} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {-\left (2 c_{1} {\mathrm e}^{x^{2}}+1\right ) {\mathrm e}^{x^{2}}}}{2 c_{1} {\mathrm e}^{x^{2}}+1} \] Verified OK.
\[ y = -\frac {\sqrt {-\left (2 c_{1} {\mathrm e}^{x^{2}}+1\right ) {\mathrm e}^{x^{2}}}}{2 c_{1} {\mathrm e}^{x^{2}}+1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{3}+{\mathrm e}^{x^{2}} y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x y^{3}}{{\mathrm e}^{x^{2}}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{3}}=-\frac {x}{{\mathrm e}^{x^{2}}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{3}}d x =\int -\frac {x}{{\mathrm e}^{x^{2}}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{2 y^{2}}=\frac {1}{2 \,{\mathrm e}^{x^{2}}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {\sqrt {-\left (2 c_{1} {\mathrm e}^{x^{2}}+1\right ) {\mathrm e}^{x^{2}}}}{2 c_{1} {\mathrm e}^{x^{2}}+1}, y=-\frac {\sqrt {-\left (2 c_{1} {\mathrm e}^{x^{2}}+1\right ) {\mathrm e}^{x^{2}}}}{2 c_{1} {\mathrm e}^{x^{2}}+1}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 33
dsolve(x*y(x)^3+exp(x^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {c_{1} -{\mathrm e}^{-x^{2}}}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {c_{1} -{\mathrm e}^{-x^{2}}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 7.124 (sec). Leaf size: 70
DSolve[x*y[x]^3+Exp[x^2]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i e^{\frac {x^2}{2}}}{\sqrt {1+2 c_1 e^{x^2}}} \\ y(x)\to \frac {i e^{\frac {x^2}{2}}}{\sqrt {1+2 c_1 e^{x^2}}} \\ y(x)\to 0 \\ \end{align*}