Internal problem ID [13292]
Internal file name [OUTPUT/12464_Wednesday_February_14_2024_02_06_27_AM_97717993/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 3. Some basics about First order equations. Additional exercises. page
63
Problem number: 3.4 f.
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]
\[ \boxed {y^{3}-25 y+y^{\prime }=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{3}+25 y}d y &= x +c_{1}\\ \frac {\ln \left (y \right )}{25}-\frac {\ln \left (y^{2}-25\right )}{50}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {5 \sqrt {\left ({\mathrm e}^{50 x +50 c_{1}}-1\right ) {\mathrm e}^{50 x +50 c_{1}}}}{{\mathrm e}^{50 x +50 c_{1}}-1}\\ &=-\frac {5 \sqrt {\left ({\mathrm e}^{50 x} c_{1}^{50}-1\right ) {\mathrm e}^{50 x} c_{1}^{50}}}{{\mathrm e}^{50 x} c_{1}^{50}-1}\\ y_2&=\frac {5 \sqrt {\left ({\mathrm e}^{50 x +50 c_{1}}-1\right ) {\mathrm e}^{50 x +50 c_{1}}}}{{\mathrm e}^{50 x +50 c_{1}}-1}\\ &=\frac {5 \sqrt {\left ({\mathrm e}^{50 x} c_{1}^{50}-1\right ) {\mathrm e}^{50 x} c_{1}^{50}}}{{\mathrm e}^{50 x} c_{1}^{50}-1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {5 \sqrt {\left ({\mathrm e}^{50 x} c_{1}^{50}-1\right ) {\mathrm e}^{50 x} c_{1}^{50}}}{{\mathrm e}^{50 x} c_{1}^{50}-1} \\ \tag{2} y &= \frac {5 \sqrt {\left ({\mathrm e}^{50 x} c_{1}^{50}-1\right ) {\mathrm e}^{50 x} c_{1}^{50}}}{{\mathrm e}^{50 x} c_{1}^{50}-1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {5 \sqrt {\left ({\mathrm e}^{50 x} c_{1}^{50}-1\right ) {\mathrm e}^{50 x} c_{1}^{50}}}{{\mathrm e}^{50 x} c_{1}^{50}-1} \] Verified OK.
\[ y = \frac {5 \sqrt {\left ({\mathrm e}^{50 x} c_{1}^{50}-1\right ) {\mathrm e}^{50 x} c_{1}^{50}}}{{\mathrm e}^{50 x} c_{1}^{50}-1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{3}-25 y+y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{3}+25 y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-y^{3}+25 y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-y^{3}+25 y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (y-5\right )}{50}-\frac {\ln \left (y+5\right )}{50}+\frac {\ln \left (y\right )}{25}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-\frac {5 \sqrt {\left ({\mathrm e}^{50 x +50 c_{1}}-1\right ) {\mathrm e}^{50 x +50 c_{1}}}}{{\mathrm e}^{50 x +50 c_{1}}-1}, y=\frac {5 \sqrt {\left ({\mathrm e}^{50 x +50 c_{1}}-1\right ) {\mathrm e}^{50 x +50 c_{1}}}}{{\mathrm e}^{50 x +50 c_{1}}-1}\right \} \end {array} \]
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: 33
dsolve(y(x)^3-25*y(x)+diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {5}{\sqrt {25 \,{\mathrm e}^{-50 x} c_{1} +1}} \\ y \left (x \right ) &= \frac {5}{\sqrt {25 \,{\mathrm e}^{-50 x} c_{1} +1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.68 (sec). Leaf size: 110
DSolve[y[x]^3-25*y[x]+y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {5 e^{25 x}}{\sqrt {e^{50 x}+e^{50 c_1}}} \\ y(x)\to \frac {5 e^{25 x}}{\sqrt {e^{50 x}+e^{50 c_1}}} \\ y(x)\to -5 \\ y(x)\to 0 \\ y(x)\to 5 \\ y(x)\to -\frac {5 e^{25 x}}{\sqrt {e^{50 x}}} \\ y(x)\to \frac {5 e^{25 x}}{\sqrt {e^{50 x}}} \\ \end{align*}