Who invented the exponential ansatz for linear differential equations with constant coefficients?

Question

Who invented using $e^{\lambda t}$ as ansatz for solving linear differential equations with constant coefficients?

Answer 1

The short answer is Euler. Some details are given in the following long quote from Hald's History of Probability and Statistics and Their Applications before 1750 (p.438):

In 1743 Euler solved the homogeneous linear differential equation of the m-th order with constant coefficients (using the same idea as de Moivre) by guessing at a particular solution of the form $y = c\,e^{rx}$ and thus deriving the characteristic equation as an algebraic equation of the $m$-th degree. He pointed out that the general solution will be a linear combination of $m$ independent particular solutions. The year before he had proved (incompletely) that a polynomial with real coefficients can be decomposed into linear and quadratic factors with real coefficients, and he could therefore assert that the characteristic equation always has $m$ roots. He found that the particular solution for a single real root has the form $y = c\,e^{rx}$, for a real root of multiplicity $k$ the form $e^{rx}$ times a polynomial in $x$ of degree $k$, and for a pair of conjugate complex roots the form $y = e^{ax}(c_1 \cos bx + c_2 \sin bx)$, where $c_1$ and $c_2$ are arbitrary constants.

In 1753 Euler solved the nonhomogeneous differential equation of the $m$-th order with constant coefficients by devising a method for reducing the order by unity and thus successively reducing the problem to the solution of a nonhomogeneous differential equation of the first order...

In the 1760s Lagrange proved that Euler's 1753 theorem also holds for a nonhomogeneous differential equation of the $m$-th order with variable coefficients so that the general solution may be expressed by means of the solution of the homogeneous equation and the solution of an adjoint equation of the first order.

The justification obviously depends on the Fundamental Theorem of Algebra, and according to the conventional wisdom Euler's reasoning for it had an irreparable gap, which was filled by Gauss. Depending on one's strictures for rigor even Gauss may not be beyond reproach, see Dunham's Euler and the Fundamental Theorem of Algebra.

The Euler-Lagrange presentation of a general nonhomogeneous solution as a sum of general homogeneous and particular solutions is recognized today as a direct consequence of linearity, this is a general form of solution to underdetermined linear systems. But of course this insight was not yet available in their time. The method for finding a particular solution which came to be known as the variation of parameters was used by Euler in a particular case already back in 1748, but was only given its final form by Lagrange in 1808-1810. Both Euler and Lagrange were solving equations related to perturbations of elliptic orbits of the planets and the Moon.