Internal problem ID [99]
Internal file name [OUTPUT/99_Sunday_June_05_2022_01_34_44_AM_997980/index.tex]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.6, Substitution methods and exact equations. Page 74
Problem number: 21.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }-y^{3}-y=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{3}+y}d y &= x +c_{1}\\ -\frac {\ln \left (y^{2}+1\right )}{2}+\ln \left (y \right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\sqrt {-\left ({\mathrm e}^{2 x +2 c_{1}}-1\right ) {\mathrm e}^{2 x +2 c_{1}}}}{{\mathrm e}^{2 x +2 c_{1}}-1}\\ &=\frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1}\\ y_2&=-\frac {\sqrt {-\left ({\mathrm e}^{2 x +2 c_{1}}-1\right ) {\mathrm e}^{2 x +2 c_{1}}}}{{\mathrm e}^{2 x +2 c_{1}}-1}\\ &=-\frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \\ \tag{2} y &= -\frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \] Verified OK.
\[ y = -\frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{3}-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{3}+y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{3}+y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{3}+y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )-\frac {\ln \left (1+y^{2}\right )}{2}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {\sqrt {-\left ({\mathrm e}^{2 x +2 c_{1}}-1\right ) {\mathrm e}^{2 x +2 c_{1}}}}{{\mathrm e}^{2 x +2 c_{1}}-1}, y=-\frac {\sqrt {-\left ({\mathrm e}^{2 x +2 c_{1}}-1\right ) {\mathrm e}^{2 x +2 c_{1}}}}{{\mathrm e}^{2 x +2 c_{1}}-1}\right \} \end {array} \]
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: 29
dsolve(diff(y(x),x) = y(x)+y(x)^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {{\mathrm e}^{-2 x} c_{1} -1}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {{\mathrm e}^{-2 x} c_{1} -1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.06 (sec). Leaf size: 57
DSolve[y'[x] == y[x]+y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i e^{x+c_1}}{\sqrt {-1+e^{2 (x+c_1)}}} \\ y(x)\to \frac {i e^{x+c_1}}{\sqrt {-1+e^{2 (x+c_1)}}} \\ \end{align*}