Solve the differential equation dy/dx + sqrt(1 + (dy/dx)^2) = e^x.

Step-by-step solution

1. 1

   Replace the derivative by a single symbol. Put $p=\dfrac{dy}{dx}$, so the equation reads

   $p+\sqrt{1+p^{2}}=e^{x}$

   Notice $y$ itself never appears: the equation determines $p$ algebraically at each $x$, and only then is one integration needed. This is the whole reason such an equation is tractable.

2. 2

   Isolate the radical before squaring. Squaring the equation as it stands would leave a cross term; isolating first avoids that:

   $\sqrt{1+p^{2}}=e^{x}-p$

   For the square root to be non-negative we need $e^{x}\ge p$, a condition to confirm at the end.

3. 3

   Square and watch $p^{2}$ cancel.

   $1+p^{2}=e^{2x}-2e^{x}p+p^{2}\ \Longrightarrow\ 1=e^{2x}-2e^{x}p$

   The quadratic terms disappear, leaving a linear equation for $p$ — this cancellation is what makes the problem solvable in closed form.

4. 4

   Solve for $p$ and recognise the hyperbolic sine.

   $2e^{x}p=e^{2x}-1\ \Longrightarrow\ p=\frac{e^{2x}-1}{2e^{x}}=\frac{e^{x}-e^{-x}}{2}=\sinh x$

5. 5

   Integrate once.

   $y=\int\sinh x\,dx=\frac{e^{x}+e^{-x}}{2}+C=\cosh x+C$

   A single arbitrary constant is correct for a first-order equation.

6. 6

   Check the solution and the squaring condition. With $p=\sinh x$: $p+\sqrt{1+\sinh^{2}x}=\sinh x+\cosh x=e^{x}$ ✓ (using $\cosh^{2}-\sinh^{2}=1$ and $\cosh x>0$). The condition $e^{x}-p=\cosh x>0$ holds for all $x$, so squaring introduced no spurious solution. Numerically at $x=2$: $\sinh 2+\cosh 2=3.6269+3.7622=7.3891=e^{2}$ ✓.