Internal problem ID [1144]
Internal file name [OUTPUT/1145_Sunday_June_05_2022_02_03_29_AM_30770682/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 (c).
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]
\[ \boxed {y^{\prime }+y^{2}+5 y=6} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{2}-5 y +6}d y &= x +c_{1}\\ -\frac {\ln \left (y -1\right )}{7}+\frac {\ln \left (y +6\right )}{7}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {6+{\mathrm e}^{7 x +7 c_{1}}}{{\mathrm e}^{7 x +7 c_{1}}-1}\\ &=\frac {6+{\mathrm e}^{7 x} c_{1}^{7}}{{\mathrm e}^{7 x} c_{1}^{7}-1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {6+{\mathrm e}^{7 x} c_{1}^{7}}{{\mathrm e}^{7 x} c_{1}^{7}-1} \\ \end{align*}
Verification of solutions
\[ y = \frac {6+{\mathrm e}^{7 x} c_{1}^{7}}{{\mathrm e}^{7 x} c_{1}^{7}-1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}+5 y=6 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2}-5 y+6 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-y^{2}-5 y+6}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-y^{2}-5 y+6}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y+6\right )}{7}-\frac {\ln \left (y-1\right )}{7}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {6+{\mathrm e}^{7 x +7 c_{1}}}{{\mathrm e}^{7 x +7 c_{1}}-1} \end {array} \]
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: 0.032 (sec). Leaf size: 23
dsolve(diff(y(x),x)+y(x)^2+5*y(x)-6=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {6+c_{1} {\mathrm e}^{7 x}}{c_{1} {\mathrm e}^{7 x}-1} \]
✓ Solution by Mathematica
Time used: 0.71 (sec). Leaf size: 46
DSolve[y'[x]+y[x]^2+5*y[x]-6==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{7 x}+6 e^{7 c_1}}{e^{7 x}-e^{7 c_1}} \\ y(x)\to -6 \\ y(x)\to 1 \\ \end{align*}