Internal problem ID [1147]
Internal file name [OUTPUT/1148_Sunday_June_05_2022_02_03_33_AM_27793055/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 (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]
\[ \boxed {6 y^{\prime }+6 y^{2}-y=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{2}+\frac {1}{6} y +\frac {1}{6}}d y &= x +c_{1}\\ -\frac {6 \ln \left (-\frac {1}{2}+y \right )}{5}+\frac {6 \ln \left (y +\frac {1}{3}\right )}{5}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {3 \,{\mathrm e}^{\frac {5 x}{6}+\frac {5 c_{1}}{6}}+2}{-6+6 \,{\mathrm e}^{\frac {5 x}{6}+\frac {5 c_{1}}{6}}}\\ &=\frac {3 \,{\mathrm e}^{\frac {5 x}{6}} c_{1} +2}{6 \,{\mathrm e}^{\frac {5 x}{6}} c_{1} -6} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {3 \,{\mathrm e}^{\frac {5 x}{6}} c_{1} +2}{6 \,{\mathrm e}^{\frac {5 x}{6}} c_{1} -6} \\ \end{align*}
Verification of solutions
\[ y = \frac {3 \,{\mathrm e}^{\frac {5 x}{6}} c_{1} +2}{6 \,{\mathrm e}^{\frac {5 x}{6}} c_{1} -6} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 6 y^{\prime }+6 y^{2}-y=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2}+\frac {y}{6}+\frac {1}{6} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-y^{2}+\frac {y}{6}+\frac {1}{6}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-y^{2}+\frac {y}{6}+\frac {1}{6}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {6 \ln \left (-1+2 y\right )}{5}+\frac {6 \ln \left (3 y+1\right )}{5}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {1+{\mathrm e}^{\frac {5 x}{6}+\frac {5 c_{1}}{6}}}{2 \,{\mathrm e}^{\frac {5 x}{6}+\frac {5 c_{1}}{6}}-3} \end {array} \]
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: 0.031 (sec). Leaf size: 24
dsolve(6*diff(y(x),x)+6*y(x)^2-y(x)-1=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {1+c_{1} {\mathrm e}^{\frac {5 x}{6}}}{2 c_{1} {\mathrm e}^{\frac {5 x}{6}}-3} \]
✓ Solution by Mathematica
Time used: 0.234 (sec). Leaf size: 56
DSolve[6*y'[x]+6*y[x]^2-y[x]-1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{5 x/6}-e^{5 c_1}}{2 e^{5 x/6}+3 e^{5 c_1}} \\ y(x)\to -\frac {1}{3} \\ y(x)\to \frac {1}{2} \\ \end{align*}