Internal problem ID [13383]
Internal file name [OUTPUT/12555_Wednesday_February_14_2024_11_58_20_PM_26996677/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 6. Simplifying through simplifiction. Additional exercises. page 114
Problem number: 6.5 (a).
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]
\[ \boxed {y^{\prime }+3 y-3 y^{3}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{3 y^{3}-3 y}d y &= x +c_{1}\\ -\frac {\ln \left (y \right )}{3}+\frac {\ln \left (y^{2}-1\right )}{6}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {1}{\sqrt {1-{\mathrm e}^{6 x +6 c_{1}}}}\\ &=\frac {1}{\sqrt {1-{\mathrm e}^{6 x} c_{1}^{6}}}\\ y_2&=-\frac {1}{\sqrt {1-{\mathrm e}^{6 x +6 c_{1}}}}\\ &=-\frac {1}{\sqrt {1-{\mathrm e}^{6 x} c_{1}^{6}}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{\sqrt {1-{\mathrm e}^{6 x} c_{1}^{6}}} \\ \tag{2} y &= -\frac {1}{\sqrt {1-{\mathrm e}^{6 x} c_{1}^{6}}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{\sqrt {1-{\mathrm e}^{6 x} c_{1}^{6}}} \] Verified OK.
\[ y = -\frac {1}{\sqrt {1-{\mathrm e}^{6 x} c_{1}^{6}}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+3 y-3 y^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-3 y+3 y^{3} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-3 y+3 y^{3}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-3 y+3 y^{3}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y-1\right )}{6}-\frac {\ln \left (y\right )}{3}+\frac {\ln \left (y+1\right )}{6}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {1}{\sqrt {1-{\mathrm e}^{6 x +6 c_{1}}}}, y=-\frac {1}{\sqrt {1-{\mathrm e}^{6 x +6 c_{1}}}}\right \} \end {array} \]
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: 0.015 (sec). Leaf size: 29
dsolve(diff(y(x),x)+3*y(x)=3*y(x)^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {{\mathrm e}^{6 x} c_{1} +1}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {{\mathrm e}^{6 x} c_{1} +1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.675 (sec). Leaf size: 58
DSolve[y'[x]+3*y[x]==3*y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{\sqrt {1+e^{6 x+2 c_1}}} \\ y(x)\to \frac {1}{\sqrt {1+e^{6 x+2 c_1}}} \\ y(x)\to -1 \\ y(x)\to 0 \\ y(x)\to 1 \\ \end{align*}