In mathematics, the exponential approach is understood to be an approach to solving a linear differential equation with constant coefficients, the inhomogeneity of which has an exponential structure. The idea is that a particulate solution of a similar shape to the inhomogeneity then also exists. Such a solution approach reduces the differential equation to a linear system of equations . The idea for this approach goes back to Leonhard Euler .

formulation

A linear differential equation is given

${\ displaystyle y ^ {(n)} (x) + \ sum _ {k = 0} ^ {n-1} c_ {k} y ^ {(k)} (x) = b (x)}$

with constant coefficients , wherein the inhomogeneity is the structure
${\ displaystyle c_ {0}, \ ldots, c_ {n-1} \ in \ mathbb {C}}$

${\ displaystyle b (x) = e ^ {(\ alpha + i \ beta) x} \ sum _ {k = 0} ^ {l} a_ {k} x ^ {k} \, \ \ alpha, \ beta \ in \ mathbb {R} \, \ a_ {0}, \ ldots, a_ {l} \ in \ mathbb {C}}$

owns. Also denote the zero order of with respect to the characteristic polynomial of the associated homogeneous equation
${\ displaystyle j \ in \ mathbb {N} _ {0}}$${\ displaystyle \ alpha + i \ beta}$

Then there is a special solution of the form
${\ displaystyle y_ {sp}}$

${\ displaystyle y_ {sp} (x) = e ^ {(\ alpha + i \ beta) x} \ sum _ {k = j} ^ {l + j} b_ {k} x ^ {k} \, \ b_ {j}, \ ldots, b_ {l + j} \ in \ mathbb {C} \.}$

example

Consider the linear differential equation

${\ displaystyle y '' (x) + y (x) = xe ^ {ix} \.}$

Now is the first order zero of the polynomial . So according to the above theorem there is a special solution of the shape
${\ displaystyle i}$${\ displaystyle \ chi (\ lambda) = \ lambda ^ {2} +1}$