Internal problem ID [12640]
Internal file name [OUTPUT/11293_Friday_November_03_2023_06_29_54_AM_30618932/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 2. The Initial Value Problem. Exercises 2.2, page 53
Problem number: 5.
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]
\[ \boxed {y^{\prime }-\sqrt {\left (y+2\right ) \left (y-1\right )}=0} \]
Integrating both sides gives \begin{align*} \int \frac {1}{\sqrt {\left (y +2\right ) \left (y -1\right )}}d y &= \int d x \\ \ln \left (\frac {1}{2}+y+\sqrt {-2+y^{2}+y}\right )&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \ln \left (\frac {1}{2}+y+\sqrt {-2+y^{2}+y}\right ) &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \ln \left (\frac {1}{2}+y+\sqrt {-2+y^{2}+y}\right ) = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {\left (y+2\right ) \left (y-1\right )}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\sqrt {\left (y+2\right ) \left (y-1\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {\left (y+2\right ) \left (y-1\right )}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {\left (y+2\right ) \left (y-1\right )}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (\frac {1}{2}+y+\sqrt {-2+y^{2}+y}\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {4 \left ({\mathrm e}^{x +c_{1}}\right )^{2}-4 \,{\mathrm e}^{x +c_{1}}+9}{8 \,{\mathrm e}^{x +c_{1}}} \end {array} \]
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: 0.0 (sec). Leaf size: 30
dsolve(diff(y(x),x)=sqrt( (y(x)+2)*(y(x)-1)),y(x), singsol=all)
\[ x +\ln \left (2\right )-\ln \left (1+2 y \left (x \right )+2 \sqrt {\left (y \left (x \right )+2\right ) \left (-1+y \left (x \right )\right )}\right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.497 (sec). Leaf size: 41
DSolve[y'[x]==Sqrt[ (y[x]+2)*(y[x]-1)],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (-e^{-x-c_1}-9 e^{x+c_1}-2\right ) \\ y(x)\to -2 \\ y(x)\to 1 \\ \end{align*}