29.4.11 problem 100

Internal problem ID [4702]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 4
Problem number : 100
Date solved : Monday, January 27, 2025 at 09:32:24 AM
CAS classification : [[_homogeneous, `class G`], _Chini]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 68

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

\[ -\frac {\ln \left (\sqrt {y \left (x \right )}\, b x +a \,x^{2}-2 y \left (x \right )\right )}{2}+\frac {b \sqrt {y \left (x \right )}\, \operatorname {arctanh}\left (\frac {b \sqrt {y \left (x \right )}+2 a x}{\sqrt {y \left (x \right ) \left (b^{2}+8 a \right )}}\right )}{\sqrt {y \left (x \right ) \left (b^{2}+8 a \right )}}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.276 (sec). Leaf size: 118

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

\[ \text {Solve}\left [\frac {b^2 \left (-\frac {2 b \text {arctanh}\left (\frac {b \left (1-\frac {4 a \sqrt {\frac {b^2 y(x)}{a^2 x^2}}}{b^2}\right )}{\sqrt {8 a+b^2}}\right )}{\sqrt {8 a+b^2}}-\log \left (b^2 \left (\sqrt {\frac {b^2 y(x)}{a^2 x^2}}+1\right )-\frac {2 b^2 y(x)}{a x^2}\right )\right )}{2 a}=\frac {b^2 \log (x)}{a}+c_1,y(x)\right ] \]