Chapter 4, Problem 65RE

a.

To determine

Find at least one nonzero function that satisfies each of these differential equations.

Answer to Problem 65RE

  $f(x)=\frac{1}{2}{x}^2+c$ .

Explanation of Solution

Given information $f\text{'}(x)=x.$

Calculation:

There are actually infinitely many functions such that derivative of the function is equal to x.

  $\begin{array}{l}\text{Let}f(x)=\frac{1}{2}{x}^2+c,\text{where}\text{c}\text{is}\text{any}\text{constant}.\\ f\text{'}(x)=\frac{1}{2}\times 2\times x+0=x.\end{array}$

So.

  $f(x)=\frac{1}{2}{x}^2+c$ .

b.

To determine

Find at least one nonzero function that satisfies each of these differential equations.

Answer to Problem 65RE

  $f(x)=C{e}^x.$

Explanation of Solution

Given information: $f\text{'}(x)=f(x).$

Calculation:

To find such function ,solve this differential equation:

  $\begin{array}{l}f\text{'}(x)=f(x).\\ \frac{d(f(x))}{dx}=f(x)\\ d(f(x))=f(x).dx\\ \frac{d(f(x))}{f(x)}=dx.\\ \text{Now}\text{integrate}\text{both}\text{sides}\text{.}\\ \ln (f(x))=x+C.\\ f(x)=C\times e^x.\\ f(x)=C{e}^x.\end{array}$

c.

To determine

Find at least one nonzero function that satisfies each of these differential equations.

Answer to Problem 65RE

  $f(x)=C{e}^{-x}.$

Explanation of Solution

Given information: $f\text{'}(x)=-f(x)$.

Calculation:

To find such function ,solve this differential equation:

  $\begin{array}{l}f\text{'}(x)=-f(x).\\ \frac{d(f(x))}{dx}=-f(x)\\ d(f(x))=-f(x).dx\\ \frac{d(f(x))}{f(x)}=-dx.\\ \text{Now}\text{integrate}\text{both}\text{sides}\text{.}\\ \ln (f(x))=-x+C.\\ f(x)=C\times e^{-x}.\\ f(x)=C{e}^{-x}.\end{array}$

d.

To determine

Find at least one nonzero function that satisfies each of these differential equations.

Answer to Problem 65RE

  $f(x)=e^x,f(x)=e^{-x}$

Explanation of Solution

Given information: $f\text{'}\text{'}(x)=f(x)$

Calculation:

Yes there some function like ,

  $\begin{array}{l}f(x)=e^x,f(x)=e^{-x}\\ f\text{'}(x)=e^x.\\ f\text{'}\text{'}(x)=e^x=f(x).\\ So,\\ f(x)=e^x.\\ f(x)=e^{-x}\\ f\text{'}(x)=-e^{-x}.\\ f\text{'}\text{'}(x)=-(-e^{-x}).\\ f\text{'}\text{'}(x)=e^{-x}=f(x).\end{array}$

e.

To determine

Find at least one nonzero function that satisfies each of these differential equations.

Answer to Problem 65RE

  $f(x)=\cos x.$

Explanation of Solution

Given information: $f\text{'}\text{'}(x)=-f(x).$

Calculation:

  $$

  $\begin{array}{l}Letf(x)=\cos x.\\ f\text{'}(x)=\frac{d\cos x}{dx}=-\sin x.\\ f\text{'}\text{'}(x)=-(\frac{d\sin x}{dx})=-(\cos x)=-f(x).\\ So,f(x)=\cos x.\end{array}$