Question

In Problems 13 and 14 determine by inspection at least one solution of the given differential equation.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{=}\mathit{y}\mathbf{\text{'}}$

Short answer

$y=c_1$and $y=c_2{e}^x$, where$c_1$and ${\text{c}}_{\text{2}}$are constants.

Step-by-step solution

Define second derivative of a function

The derivative of the derivative of a function f is known as the second derivative, or second order derivative, in calculus.

So, the second derivative, or the rate of change of speed with respect to time, can be used to determine the variation in speed of the car (the second derivative of distance travelled with respect to the time).

Determine the solution of the given differential equation

Find a function that has the same second derivative as the first. The solutions for the preceding differential equation are $y=c_1$or $y=c_2{e}^x$, where $c_1$and ${\text{c}}_{\text{2}}$are constants.

As already known from the previous section.

Let check this:

As the constant function, $y=c_1$then $y\text{'}=0$and $y\text{'}\text{'}=0$.

Therefore,.

Also, as the constant function, $y=c_2{e}^x$then and $y\text{'}\text{'}=c_2{e}^x$.

Therefore, $y\text{'}\text{'}=y\text{'}$.

Hence, $y=c_1$and $y=c_2{e}^x$, where $c_1$and ${\text{c}}_{\text{2}}$are constants.