Chapter 10.3, Problem 35E

(a)

To determine

Whether it is possible to find a formula for the function y that does not involve any integrals.

Answer to Problem 35E

No, we cannot to find a formula for the function y that does not involve any integrals.

Explanation of Solution

Given information:

  $\frac{dy}{dx}=e^{-x^2}$

And

  $y=2$

When

  $x=0$

Since

  $e^{-x^2}$ is a non − differentiable function.

Thus,

The differential equation cannot be solved for the formula of y.

However,

Only way to solve the differential equation is to write the series representation of $e^{-x^2}$ and integrate both sides with respect to x to get y in terms of x.

Then

Eliminate the constant term with the initial condition.

Therefore,

It is not possible to find a formula for the function y that does not involve any integrals.

(b)

To determine

Whether it is possible to represent y by a power series.

Answer to Problem 35E

Yes, we can represent y with a power series.

Explanation of Solution

Given information:

  $\frac{dy}{dx}=e^{-x^2}$

And

  $y=2$

When

  $x=0$

We have

  $\frac{dy}{dx}=e^{-x^2}={\displaystyle {\sum }_{n=0}^{\infty }\frac{{\left(-x^2\right)}^n}{n!}}={\displaystyle {\sum }_{n=0}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n}}{n!}}$

Integrate:

  $y={\displaystyle \int e^{-x^2}dx}={\displaystyle \int {\displaystyle {\sum }_{n=0}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n}}{n!}dx}}={\displaystyle {\sum }_{n=0}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n+1}}{n!\left(2n+1\right)}}+C$

But

  $2=C$

On putting

  $y\left(0\right)=2$

  $y={\displaystyle {\sum }_{n=0}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n+1}}{n!\left(2n+1\right)}}+2$

Therefore,

y can be represented by power series.

(c)

To determine

Values of x for the power series actually equal the function y.

Answer to Problem 35E

Power series equals y for only $x=0$ .

Explanation of Solution

Given information:

  $\frac{dy}{dx}=e^{-x^2}$

And

  $y=2$

When

  $x=0$

For some finite value of n,

The power series equals y when the error term is zero.

That occurs only for $x=0$ .

According to the given initial condition,

For which

  $y=2$

Therefore,

Power series equals y for only $x=0$ .