Section 6.3 of [Strang] p. 242 calls *y*' = *ky* "the most important differential equation in applied mathematics". In the field of population ecology it is known as the *Malthusian growth model* or *exponential law*, and *C*, *k*, and *t* correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model); and in finance, the model appears in a similar role in *continuous compounding* with *C* as the initial amount of money. In *exponential decay* models, *k* is often expressed as the negative of a positive constant λ.

Here *y*' is given as (*S*D*Y*), *C* as *c*, and *ky* as ((*S* × {*K*}) ∘_{f} · *Y*). (*S* × {*K*}) is the constant function that maps any real or complex input to *k* and ∘_{f} · is multiplication as a function operation.

The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " *C* is the *initial value* of *y*, and *k* is the *proportionality constant*. *Exponential growth* occurs when *k* > 0, and *exponential decay* occurs when *k* < 0."); its proof here closely follows the proof of *y*' = *y* in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case.

Statements for this and expgrowthi 24096 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

⊢ (*φ* → *S* ∈ {ℝ, ℂ}) &

⊢ (*φ* → *K* ∈ ℂ) &

⊢ (*φ* → *Y*:*S*–→ℂ) &

⊢ (*φ* → dom ( *S*D*Y*) = *S*) ⇒

⊢ (*φ* → ((*S*D*Y*) = ((*S* × {*K*}) ∘_{f} · *Y*) ↔ ∃*c* ∈ ℂ *Y* = (*t* ∈ *S* ↦ (*c* · (exp ‘(*K* · *t*))))))

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⊢ (((*A* ∈ *V* ∧ *F*:*A*–→ℂ) ∧ (*G*:*A*–→(ℂ ∖ {0}) ∧ *H*:*A*–→(ℂ ∖ {0}))) → (*F* ∘_{f} / (*G* ∘_{f} / *H*)) = ((*F* ∘_{f} · *H*) ∘_{f} / *G*))

]]>*S* ∈ {ℝ, ℂ} → ℝ ⊆ *S*)

]]>*A* ⊆ (*B* ∖ *A*) ↔ *A* = ∅)

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⊢ (*φ* → *A* ∈ *V*) &

⊢ (*φ* → *F*:*A*–→*T*) &

⊢ (*φ* → *G*:*A*–→*S*) &

⊢ (*φ* → *H*:*A*–→*S*) &

⊢ ((*φ* ∧ (*x* ∈ *T* ∧ *y* ∈ *S* ∧ *z* ∈ *S*)) → ((*x**R**y*) = (*x**R**z*) ↔ *y* = *z*)) ⇒

⊢ (*φ* → ((*F* ∘_{f} *R**G*) = (*F* ∘_{f} *R**H*) ↔ *G* = *H*))

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⊢ (((*A* ∈ *V* ∧ *F*:*A*–→ℂ) ∧ (*G*:*A*–→ℂ ∧ *H*:*A*–→ℂ)) → (*F* ∘_{f} · (*G* ∘_{f} · *H*)) = (*G* ∘_{f} · (*F* ∘_{f} · *H*)))

]]>*S* ∈ {ℝ, ℂ} → (*S*D(exp ↾ *S*)) = (exp ↾ *S*))

]]>*S* analog of dvconst 16969 and dveq0 17038. Corresponds to integration formula "∫0 d*x* = *C* " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)

⊢ (*φ* → *S* ∈ {ℝ, ℂ}) &

⊢ (*φ* → *Y*:*S*–→ℂ) &

⊢ (*φ* → dom ( *S*D*Y*) = *S*) ⇒

⊢ (*φ* → ((*S*D*Y*) = (*S* × {0}) ↔ ∃*c* ∈ ℂ *Y* = (*S* × {*c*})))

]]>*S* ∈ {ℝ, ℂ} → (*S*D( I ↾ *S*)) = (*S* × {1}))

]]>*S* is ℝ. (Contributed by Steve Rodriguez, 11-Nov-2015.)

⊢ ((*S* ∈ {ℝ, ℂ} ∧ *A* ∈ ℂ) → (*S*D(*S* × {*A*})) = (*S* × {0}))]]>