Internal problem ID [13354]
Internal file name [OUTPUT/12526_Wednesday_February_14_2024_11_54_41_PM_18544785/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 5. LINEAR FIRST ORDER EQUATIONS. Additional exercises. page
103
Problem number: 5.1 (i).
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]
\[ \boxed {y^{\prime }+4 y-y^{3}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{3}-4 y}d y &= x +c_{1}\\ -\frac {\ln \left (y \right )}{4}+\frac {\ln \left (y^{2}-4\right )}{8}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {2}{\sqrt {1-{\mathrm e}^{8 x +8 c_{1}}}}\\ &=-\frac {2}{\sqrt {1-{\mathrm e}^{8 x} c_{1}^{8}}}\\ y_2&=\frac {2}{\sqrt {1-{\mathrm e}^{8 x +8 c_{1}}}}\\ &=\frac {2}{\sqrt {1-{\mathrm e}^{8 x} c_{1}^{8}}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {2}{\sqrt {1-{\mathrm e}^{8 x} c_{1}^{8}}} \\ \tag{2} y &= \frac {2}{\sqrt {1-{\mathrm e}^{8 x} c_{1}^{8}}} \\ \end{align*}
Verification of solutions
\[ y = -\frac {2}{\sqrt {1-{\mathrm e}^{8 x} c_{1}^{8}}} \] Verified OK.
\[ y = \frac {2}{\sqrt {1-{\mathrm e}^{8 x} c_{1}^{8}}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+4 y-y^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-4 y+y^{3} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-4 y+y^{3}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-4 y+y^{3}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y-2\right )}{8}+\frac {\ln \left (y+2\right )}{8}-\frac {\ln \left (y\right )}{4}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-\frac {2}{\sqrt {1-{\mathrm e}^{8 x +8 c_{1}}}}, y=\frac {2}{\sqrt {1-{\mathrm e}^{8 x +8 c_{1}}}}\right \} \end {array} \]
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: 0.016 (sec). Leaf size: 33
dsolve(diff(y(x),x)+4*y(x)=y(x)^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {2}{\sqrt {4 \,{\mathrm e}^{8 x} c_{1} +1}} \\ y \left (x \right ) &= \frac {2}{\sqrt {4 \,{\mathrm e}^{8 x} c_{1} +1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.728 (sec). Leaf size: 56
DSolve[y'[x]+4*y[x]==y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2}{\sqrt {1+e^{8 (x+c_1)}}} \\ y(x)\to \frac {2}{\sqrt {1+e^{8 (x+c_1)}}} \\ y(x)\to -2 \\ y(x)\to 0 \\ y(x)\to 2 \\ \end{align*}