Internal problem ID [1143]
Internal file name [OUTPUT/1144_Sunday_June_05_2022_02_03_28_AM_82082756/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page
253
Problem number: 38 part (b).
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 -1\right )-\ln \left (y -2\right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {2 \,{\mathrm e}^{x +c_{1}}-1}{{\mathrm e}^{x +c_{1}}-1}\\ &=\frac {2 c_{1} {\mathrm e}^{x}-1}{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}-1}{c_{1} {\mathrm e}^{x}-1} \\ \end{align*}
Verification of solutions
\[ y = \frac {2 c_{1} {\mathrm e}^{x}-1}{c_{1} {\mathrm e}^{x}-1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}-3 y=-2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2}+3 y-2 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-y^{2}+3 y-2}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-y^{2}+3 y-2}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (y-2\right )+\ln \left (y-1\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {2 \,{\mathrm e}^{x +c_{1}}-1}{{\mathrm e}^{x +c_{1}}-1} \end {array} \]
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: 0.016 (sec). Leaf size: 20
dsolve(diff(y(x),x)+y(x)^2-3*y(x)+2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 \,{\mathrm e}^{x} c_{1} -1}{{\mathrm e}^{x} c_{1} -1} \]
✓ Solution by Mathematica
Time used: 0.93 (sec). Leaf size: 40
DSolve[y'[x]+y[x]^2-3*y[x]+2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 e^x-e^{c_1}}{e^x-e^{c_1}} \\ y(x)\to 1 \\ y(x)\to 2 \\ \end{align*}