Internal problem ID [3361]
Internal file name [OUTPUT/2853_Sunday_June_05_2022_08_41_50_AM_19164153/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 4
Problem number: 103.
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]
\[ \boxed {y^{\prime }-\sqrt {a +b y^{2}}=0} \]
Integrating both sides gives \begin{align*} \int \frac {1}{\sqrt {b \,y^{2}+a}}d y &= \int d x \\ \frac {\ln \left (\sqrt {b}\, y+\sqrt {a +b y^{2}}\right )}{\sqrt {b}}&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \frac {\ln \left (\sqrt {b}\, y+\sqrt {a +b y^{2}}\right )}{\sqrt {b}} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \frac {\ln \left (\sqrt {b}\, y+\sqrt {a +b y^{2}}\right )}{\sqrt {b}} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {a +b y^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\sqrt {a +b y^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {a +b y^{2}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {a +b y^{2}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (\sqrt {b}\, y+\sqrt {a +b y^{2}}\right )}{\sqrt {b}}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left ({\mathrm e}^{\sqrt {b}\, c_{1} +x \sqrt {b}}\right )^{2}-a}{2 \,{\mathrm e}^{\sqrt {b}\, c_{1} +x \sqrt {b}} \sqrt {b}} \end {array} \]
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: 35
dsolve(diff(y(x),x) = sqrt(a+b*y(x)^2),y(x), singsol=all)
\[ \frac {\left (c_{1} +x \right ) \sqrt {b}-\ln \left (y \left (x \right ) \sqrt {b}+\sqrt {a +b y \left (x \right )^{2}}\right )}{\sqrt {b}} = 0 \]
✓ Solution by Mathematica
Time used: 60.157 (sec). Leaf size: 82
DSolve[y'[x]==Sqrt[a+b y[x]^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {a} \tanh \left (\sqrt {b} (x+c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x+c_1)\right )}} \\ y(x)\to \frac {\sqrt {a} \tanh \left (\sqrt {b} (x+c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x+c_1)\right )}} \\ \end{align*}