Internal problem ID [1145]
Internal file name [OUTPUT/1146_Sunday_June_05_2022_02_03_31_AM_37228273/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 (d).
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]
\[ \boxed {y^{\prime }+y^{2}+8 y=-7} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{2}-8 y -7}d y &= x +c_{1}\\ -\frac {\ln \left (y +1\right )}{6}+\frac {\ln \left (7+y \right )}{6}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {-7+{\mathrm e}^{6 x +6 c_{1}}}{{\mathrm e}^{6 x +6 c_{1}}-1}\\ &=-\frac {-7+{\mathrm e}^{6 x} c_{1}^{6}}{{\mathrm e}^{6 x} c_{1}^{6}-1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {-7+{\mathrm e}^{6 x} c_{1}^{6}}{{\mathrm e}^{6 x} c_{1}^{6}-1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {-7+{\mathrm e}^{6 x} c_{1}^{6}}{{\mathrm e}^{6 x} c_{1}^{6}-1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}+8 y=-7 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2}-8 y-7 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-y^{2}-8 y-7}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-y^{2}-8 y-7}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (1+y\right )}{6}+\frac {\ln \left (7+y\right )}{6}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {-7+{\mathrm e}^{6 x +6 c_{1}}}{{\mathrm e}^{6 x +6 c_{1}}-1} \end {array} \]
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: 24
dsolve(diff(y(x),x)+y(x)^2+8*y(x)+7=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {7-c_{1} {\mathrm e}^{6 x}}{c_{1} {\mathrm e}^{6 x}-1} \]
✓ Solution by Mathematica
Time used: 0.655 (sec). Leaf size: 47
DSolve[y'[x]+y[x]^2+8*y[x]+7==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{6 x}-7 e^{6 c_1}}{e^{6 x}-e^{6 c_1}} \\ y(x)\to -7 \\ y(x)\to -1 \\ \end{align*}