Internal problem ID [3357]
Internal file name [OUTPUT/2849_Sunday_June_05_2022_08_41_45_AM_99150472/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 4
Problem number: 99.
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]
\[ \boxed {y^{\prime }-y b -\sqrt {\operatorname {A0} +\operatorname {B0} y}=a} \]
Integrating both sides gives \begin {align*} \int \frac {1}{a +b y +\sqrt {\operatorname {B0} y +\operatorname {A0}}}d y &= \int {dx}\\ \int _{}^{y}\frac {1}{a +b \textit {\_a} +\sqrt {\operatorname {B0} \textit {\_a} +\operatorname {A0}}}d \textit {\_a}&= x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{a +b \textit {\_a} +\sqrt {\operatorname {B0} \textit {\_a} +\operatorname {A0}}}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {1}{a +b \textit {\_a} +\sqrt {\operatorname {B0} \textit {\_a} +\operatorname {A0}}}d \textit {\_a} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y b -\sqrt {\mathit {A0} +\mathit {B0} y}=a \\ \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 }=a +y b +\sqrt {\mathit {A0} +\mathit {B0} y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{a +y b +\sqrt {\mathit {A0} +\mathit {B0} y}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{a +y b +\sqrt {\mathit {A0} +\mathit {B0} y}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (-b \left (\mathit {A0} +\mathit {B0} y\right )+\mathit {A0} b +\sqrt {\mathit {A0} +\mathit {B0} y}\, \mathit {B0} -a \mathit {B0} \right )}{2 b}+\frac {\mathit {B0} \arctan \left (\frac {-2 b \sqrt {\mathit {A0} +\mathit {B0} y}+\mathit {B0}}{\sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}\right )}{b \sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}+\frac {\ln \left (-b \left (\mathit {A0} +\mathit {B0} y\right )+\mathit {A0} b -\sqrt {\mathit {A0} +\mathit {B0} y}\, \mathit {B0} -a \mathit {B0} \right )}{2 b}+\frac {\mathit {B0} \arctan \left (\frac {-2 b \sqrt {\mathit {A0} +\mathit {B0} y}-\mathit {B0}}{\sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}\right )}{b \sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}+\frac {\ln \left (-y^{2} b^{2}-2 y a b +\mathit {B0} y-a^{2}+\mathit {A0} \right )}{2 b}-\frac {\arctan \left (\frac {-2 y b^{2}-2 a b +\mathit {B0}}{\sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}\right ) \mathit {B0}}{b \sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}=x +c_{1} \\ \bullet & {} & \textrm {Convert}\hspace {3pt} \arctan \mathrm {to \esapos ln\esapos } \\ {} & {} & -\frac {\ln \left (-b \left (\mathit {A0} +\mathit {B0} y\right )+\mathit {A0} b +\sqrt {\mathit {A0} +\mathit {B0} y}\, \mathit {B0} -a \mathit {B0} \right )}{2 b}+\frac {\mathrm {I} \mathit {B0} \left (\ln \left (1-\frac {\mathrm {I} \left (-2 b \sqrt {\mathit {A0} +\mathit {B0} y}+\mathit {B0} \right )}{\sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}\right )-\ln \left (1+\frac {\mathrm {I} \left (-2 b \sqrt {\mathit {A0} +\mathit {B0} y}+\mathit {B0} \right )}{\sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}\right )\right )}{2 b \sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}+\frac {\ln \left (-b \left (\mathit {A0} +\mathit {B0} y\right )+\mathit {A0} b -\sqrt {\mathit {A0} +\mathit {B0} y}\, \mathit {B0} -a \mathit {B0} \right )}{2 b}+\frac {\mathrm {I} \mathit {B0} \left (\ln \left (1+\frac {\mathrm {I} \left (2 b \sqrt {\mathit {A0} +\mathit {B0} y}+\mathit {B0} \right )}{\sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}\right )-\ln \left (1-\frac {\mathrm {I} \left (2 b \sqrt {\mathit {A0} +\mathit {B0} y}+\mathit {B0} \right )}{\sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}\right )\right )}{2 b \sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}+\frac {\ln \left (-y^{2} b^{2}-2 y a b +\mathit {B0} y-a^{2}+\mathit {A0} \right )}{2 b}-\frac {\mathrm {I} \left (\ln \left (1-\frac {\mathrm {I} \left (-2 y b^{2}-2 a b +\mathit {B0} \right )}{\sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}\right )-\ln \left (1+\frac {\mathrm {I} \left (-2 y b^{2}-2 a b +\mathit {B0} \right )}{\sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}\right )\right ) \mathit {B0}}{2 b \sqrt {-4 \mathit {A0} \,b^{2}+4 a b \mathit {B0} -\mathit {B0}^{2}}}=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: 27
dsolve(diff(y(x),x) = a+b*y(x)+sqrt(A0+B0*y(x)),y(x), singsol=all)
\[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{a +b \textit {\_a} +\sqrt {\operatorname {B0} \textit {\_a} +\operatorname {A0}}}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.693 (sec). Leaf size: 172
DSolve[y'[x]==a+b y[x]+Sqrt[A0+B0 y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\log \left (-\text {B0} \left (\sqrt {\text {$\#$1} \text {B0}+\text {A0}}+\text {$\#$1} b+a\right )\right )-\frac {2 \text {B0} \arctan \left (\frac {2 b \sqrt {\text {$\#$1} \text {B0}+\text {A0}}+\text {B0}}{\sqrt {\text {B0} (4 a b-\text {B0})-4 \text {A0} b^2}}\right )}{\sqrt {\text {B0} (4 a b-\text {B0})-4 \text {A0} b^2}}}{b}\&\right ][x+c_1] \\ y(x)\to -\frac {\sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2}+2 a b-\text {B0}}{2 b^2} \\ y(x)\to \frac {\sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2}-2 a b+\text {B0}}{2 b^2} \\ \end{align*}