Internal problem ID [13323]
Internal file name [OUTPUT/12495_Wednesday_February_14_2024_02_06_51_AM_70001529/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page
90
Problem number: 4.6 (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]
\[ \boxed {y^{\prime }-200 y+2 y^{2}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-2 y^{2}+200 y}d y &= x +c_{1}\\ \frac {\ln \left (y \right )}{200}-\frac {\ln \left (y -100\right )}{200}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {100 \,{\mathrm e}^{200 x +200 c_{1}}}{-1+{\mathrm e}^{200 x +200 c_{1}}}\\ &=\frac {100 \,{\mathrm e}^{200 x} c_{1}^{200}}{-1+{\mathrm e}^{200 x} c_{1}^{200}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {100 \,{\mathrm e}^{200 x} c_{1}^{200}}{-1+{\mathrm e}^{200 x} c_{1}^{200}} \\ \end{align*}
Verification of solutions
\[ y = \frac {100 \,{\mathrm e}^{200 x} c_{1}^{200}}{-1+{\mathrm e}^{200 x} c_{1}^{200}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-200 y+2 y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=200 y-2 y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{200 y-2 y^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{200 y-2 y^{2}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y\right )}{200}-\frac {\ln \left (y-100\right )}{200}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {100 \,{\mathrm e}^{200 x +200 c_{1}}}{-1+{\mathrm e}^{200 x +200 c_{1}}} \end {array} \]
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: 17
dsolve(diff(y(x),x)=200*y(x)-2*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {100}{1+100 \,{\mathrm e}^{-200 x} c_{1}} \]
✓ Solution by Mathematica
Time used: 0.285 (sec). Leaf size: 36
DSolve[y'[x]==200*y[x]-2*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {100 e^{200 x}}{e^{200 x}+e^{100 c_1}} \\ y(x)\to 0 \\ y(x)\to 100 \\ \end{align*}