29.4.10 problem 99

Internal problem ID [4701]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 4
Problem number : 99
Date solved : Monday, January 27, 2025 at 09:32:18 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=a +b y+\sqrt {\operatorname {A0} +\operatorname {B0} y} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 27

dsolve(diff(y(x),x) = a+b*y(x)+sqrt(A0+B0*y(x)),y(x), singsol=all)
 

\[ x -\int _{}^{y \left (x \right )}\frac {1}{a +b \textit {\_a} +\sqrt {\operatorname {B0} \textit {\_a} +\operatorname {A0}}}d \textit {\_a} +c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.620 (sec). Leaf size: 199

DSolve[D[y[x],x]==a+b y[x]+Sqrt[A0+B0 y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {2 \text {B0} \sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2} \text {arctanh}\left (\frac {2 b \sqrt {\text {$\#$1} \text {B0}+\text {A0}}+\text {B0}}{\sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2}}\right )}{b \left (\text {B0}^2-4 b (a \text {B0}-\text {A0} b)\right )}+\frac {\log \left (-b (\text {$\#$1} \text {B0}+\text {A0})-\text {B0} \sqrt {\text {$\#$1} \text {B0}+\text {A0}}-a \text {B0}+\text {A0} b\right )}{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*}