Internal problem ID [3362]
Internal file name [OUTPUT/2854_Sunday_June_05_2022_08_41_51_AM_84143893/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 4
Problem number: 104.
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]
\[ \boxed {y^{\prime }-y \sqrt {y b +a}=0} \]
Integrating both sides gives \begin{align*} \int \frac {1}{y \sqrt {b y +a}}d y &= \int d x \\ -\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {y b +a}}{\sqrt {a}}\right )}{\sqrt {a}}&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} -\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {y b +a}}{\sqrt {a}}\right )}{\sqrt {a}} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ -\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {y b +a}}{\sqrt {a}}\right )}{\sqrt {a}} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y \sqrt {y b +a}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y \sqrt {y b +a} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y \sqrt {y b +a}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y \sqrt {y b +a}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {2 \,\mathrm {arctanh}\left (\frac {\sqrt {y b +a}}{\sqrt {a}}\right )}{\sqrt {a}}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {a \left (\tanh \left (\frac {c_{1} \sqrt {a}}{2}+\frac {x \sqrt {a}}{2}\right )^{2}-1\right )}{b} \end {array} \]
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: 24
dsolve(diff(y(x),x) = y(x)*sqrt(a+b*y(x)),y(x), singsol=all)
\[ x +\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b y \left (x \right )}}{\sqrt {a}}\right )}{\sqrt {a}}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 19.061 (sec). Leaf size: 42
DSolve[y'[x]==y[x] Sqrt[a+b y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {a \text {sech}^2\left (\frac {1}{2} \sqrt {a} (x+c_1)\right )}{b} \\ y(x)\to 0 \\ y(x)\to -\frac {a}{b} \\ \end{align*}