Internal problem ID [12968]
Internal file name [OUTPUT/11621_Tuesday_November_07_2023_11_52_12_PM_45223112/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.6 page 89
Problem number: 6.
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]
\[ \boxed {y^{\prime }-\frac {1}{y-2}=0} \]
Integrating both sides gives \begin {align*} \int \left (y -2\right )d y &= t +c_{1}\\ \frac {y \left (y -4\right )}{2}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=2-\sqrt {4+2 t +2 c_{1}}\\ y_2&=2+\sqrt {4+2 t +2 c_{1}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2-\sqrt {4+2 t +2 c_{1}} \\ \tag{2} y &= 2+\sqrt {4+2 t +2 c_{1}} \\ \end{align*}
Verification of solutions
\[ y = 2-\sqrt {4+2 t +2 c_{1}} \] Verified OK.
\[ y = 2+\sqrt {4+2 t +2 c_{1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {1}{y-2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{y-2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } \left (y-2\right )=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int y^{\prime } \left (y-2\right )d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}-2 y=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=2-\sqrt {4+2 t +2 c_{1}}, y=2+\sqrt {4+2 t +2 c_{1}}\right \} \end {array} \]
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: 33
dsolve(diff(y(t),t)=1/(y(t)-2),y(t), singsol=all)
\begin{align*} y \left (t \right ) &= 2-\sqrt {4+2 t +2 c_{1}} \\ y \left (t \right ) &= 2+\sqrt {4+2 t +2 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.145 (sec). Leaf size: 44
DSolve[y'[t]==1/(y[t]-2),y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to 2-\sqrt {2} \sqrt {t+2+c_1} \\ y(t)\to 2+\sqrt {2} \sqrt {t+2+c_1} \\ \end{align*}