Internal problem ID [14189]
Internal file name [OUTPUT/13870_Saturday_March_09_2024_03_56_29_PM_54095803/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number: 32.
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]
\[ \boxed {y^{\prime }-y^{2}+3 y=2} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{2}-3 y +2}d y &= x +c_{1}\\ \ln \left (y -2\right )-\ln \left (y -1\right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {-2+{\mathrm e}^{x +c_{1}}}{{\mathrm e}^{x +c_{1}}-1}\\ &=\frac {-2+c_{1} {\mathrm e}^{x}}{c_{1} {\mathrm e}^{x}-1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-2+c_{1} {\mathrm e}^{x}}{c_{1} {\mathrm e}^{x}-1} \\ \end{align*}
Verification of solutions
\[ y = \frac {-2+c_{1} {\mathrm e}^{x}}{c_{1} {\mathrm e}^{x}-1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}+3 y=2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}-3 y+2 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}-3 y+2}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}-3 y+2}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y-2\right )-\ln \left (y-1\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {-2+{\mathrm e}^{x +c_{1}}}{{\mathrm e}^{x +c_{1}}-1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 19
dsolve(diff(y(x),x)=y(x)^2-3*y(x)+2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-2+c_{1} {\mathrm e}^{x}}{-1+c_{1} {\mathrm e}^{x}} \]
✓ Solution by Mathematica
Time used: 0.873 (sec). Leaf size: 34
DSolve[y'[x]==y[x]^2-3*y[x]+2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2+e^{x+c_1}}{-1+e^{x+c_1}} \\ y(x)\to 1 \\ y(x)\to 2 \\ \end{align*}