4.17.13 \(y'(x)^2-y(x) y'(x)+e^x=0\)

ODE
\[ y'(x)^2-y(x) y'(x)+e^x=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries]]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 0.196364 (sec), leaf count = 137

\[\left \{\left \{y(x)\to -\frac {2 i e^{x/2}}{\sqrt {\tanh ^2\left (\frac {1}{2} \left (x-c_1\right )\right )-1}}\right \},\left \{y(x)\to \frac {2 i e^{x/2}}{\sqrt {\tanh ^2\left (\frac {1}{2} \left (x-c_1\right )\right )-1}}\right \},\left \{y(x)\to -\frac {2 i e^{x/2}}{\sqrt {\tanh ^2\left (\frac {1}{2} \left (c_1-x\right )\right )-1}}\right \},\left \{y(x)\to \frac {2 i e^{x/2}}{\sqrt {\tanh ^2\left (\frac {1}{2} \left (c_1-x\right )\right )-1}}\right \}\right \}\]

Maple
cpu = 0.269 (sec), leaf count = 23

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-4\,{{\rm e}^{x}}=0,y \relax (x ) ={{\it \_C1}}^{-1}+{\it \_C1}\,{{\rm e}^{x}} \right \} \] Mathematica raw input

DSolve[E^x - y[x]*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> ((-2*I)*E^(x/2))/Sqrt[-1 + Tanh[(x - C[1])/2]^2]}, {y[x] -> ((2*I)*E^(
x/2))/Sqrt[-1 + Tanh[(x - C[1])/2]^2]}, {y[x] -> ((-2*I)*E^(x/2))/Sqrt[-1 + Tanh
[(-x + C[1])/2]^2]}, {y[x] -> ((2*I)*E^(x/2))/Sqrt[-1 + Tanh[(-x + C[1])/2]^2]}}

Maple raw input

dsolve(diff(y(x),x)^2-y(x)*diff(y(x),x)+exp(x) = 0, y(x),'implicit')

Maple raw output

y(x)^2-4*exp(x) = 0, y(x) = 1/_C1+_C1*exp(x)