# (p.159) Appendix B Differential Equations

# (p.159) Appendix B Differential Equations

# B.1 The Simplest Differential Equation

Suppose *f*:ℝ → ℝ is a continuous function. Let *x* _{0} and *A* be given real numbers. If we want to find a differentiable function *y*:ℝ → ℝ such that

*y*′ =

*f*with initial condition

*y*(

*x*

_{0}) =

*A*. This equation is solved using the Fundamental Theorem of Calculus.

Theorem 27Fundamental Theorem of CalculusIf two differentiable functions g(x)and h(x)have the same derivative then they differ by a constant.

To solve (B.1) consider $\psi \left(x\right)=A+{\int}_{{x}_{0}}^{x}f\left(y\right)\text{d}y$. Then note that ψ(*x* _{0}) = *A* and ψ′(*x*) = *f*(*x*) for every *x* since *f*(·) is continuous. Suppose now that *y*(·) solves (B.1). Then since *y*′(*x*) = *f*(*x*) = ψ′(*x*) we have by the fundamental theorem of calculus that ψ(*x*) = *y*(*x*) + *C* for some constant *C*. Choosing *x* = *x* _{0} we see that *C* = 0.

# B.2 Integrating Factor

The integrating factor is used to solve linear differential equations. It is used several times in this book. It can be applied to equations of the following form:

*bp*)′ =

*b*′

*p*+

*bp*′. Suppose we multiply the left-hand side by a function

*P*(

*x*). It becomes

*P*(

*x*) such that

*P*′(

*x*) =

*P*(

*x*)

*Q*(

*x*) (this

*P*is called an integrating factor) we transform the left-hand side into the derivative of the product

*b*(

*x*)

*P*(

*x*):

*P*such that

*P*′ =

*PQ*is easy. Note that

*P*(

*x*) = e

^{∫Q(x) dx}is an integrating factor. And our differential equation has a solution: