4.3.8 \(y'(x)=\text {R1}\left (x,\sqrt {X}\right ) \text {R2}\left (y(x),\sqrt {Y}\right )\)

ODE
\[ y'(x)=\text {R1}\left (x,\sqrt {X}\right ) \text {R2}\left (y(x),\sqrt {Y}\right ) \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.10644 (sec), leaf count = 41

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{\text {R2}\left (K[1],\sqrt {Y}\right )} \, dK[1]\& \right ]\left [\int _1^x \text {R1}\left (K[2],\sqrt {X}\right ) \, dK[2]+c_1\right ]\right \}\right \}\]

Maple
cpu = 0.036 (sec), leaf count = 26

\[ \left \{ \int \!{\it R1} \left (x,\sqrt {X} \right ) \,{\rm d}x-\int ^{y \relax (x ) }\! \left ({\it R2} \left ({\it \_a},\sqrt {Y} \right ) \right ) ^{-1}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y'[x] == R1[x, Sqrt[X]]*R2[y[x], Sqrt[Y]],y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[Integrate[R2[K[1], Sqrt[Y]]^(-1), {K[1], 1, #1}] & ][C
[1] + Integrate[R1[K[2], Sqrt[X]], {K[2], 1, x}]]}}

Maple raw input

dsolve(diff(y(x),x) = R1(x,X^(1/2))*R2(y(x),Y^(1/2)), y(x),'implicit')

Maple raw output

Int(R1(x,X^(1/2)),x)-Intat(1/R2(_a,Y^(1/2)),_a = y(x))+_C1 = 0