Internal problem ID [540]
Internal file name [OUTPUT/540_Sunday_June_05_2022_01_43_57_AM_7870853/index.tex]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.5. Page 88
Problem number: 11.
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]
\[ \boxed {y^{\prime }+b \sqrt {y}-a y=0} \]
Integrating both sides gives \begin{align*} \int \frac {1}{-b \sqrt {y}+a y}d y &= \int d t \\ \frac {2 \ln \left (a \sqrt {y}-b \right )}{a}&=t +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \frac {2 \ln \left (a \sqrt {y}-b \right )}{a} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ \frac {2 \ln \left (a \sqrt {y}-b \right )}{a} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+b \sqrt {y}-a y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-b \sqrt {y}+a y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-b \sqrt {y}+a y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{-b \sqrt {y}+a y}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y a^{2}-b^{2}\right )}{a}-\frac {\ln \left (a \sqrt {y}+b \right )}{a}+\frac {\ln \left (a \sqrt {y}-b \right )}{a}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {2 b \left (b -{\mathrm e}^{\frac {1}{2} c_{1} a +\frac {1}{2} t a}\right )-b^{2}+{\mathrm e}^{c_{1} a +t a}}{a^{2}}, y=\frac {2 b \left (b +{\mathrm e}^{\frac {1}{2} c_{1} a +\frac {1}{2} t a}\right )-b^{2}+{\mathrm e}^{c_{1} a +t a}}{a^{2}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 26
dsolve(diff(y(t),t) = -b*y(t)^(1/2)+a*y(t),y(t), singsol=all)
\[ \frac {-{\mathrm e}^{\frac {a t}{2}} c_{1} a +\sqrt {y \left (t \right )}\, a -b}{a} = 0 \]
✓ Solution by Mathematica
Time used: 0.844 (sec). Leaf size: 55
DSolve[y'[t] == -b*y[t]^(1/2)+a*y[t],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {e^{-a c_1} \left (e^{\frac {a t}{2}}-b e^{\frac {a c_1}{2}}\right ){}^2}{a^2} \\ y(t)\to 0 \\ y(t)\to \frac {b^2}{a^2} \\ \end{align*}