Question

In Problems 1 and 2 Fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol ${\mathit{c}}_{\mathbf{1}}$and has the form $\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}\mathbf{=}\mathit{f}\left(x,y\right)$. The symbol ${\mathit{c}}_{\mathbf{1}}$represents a constant.

$\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{\mathbf{10}\mathbf{x}}\mathbf{=}\mathit{\_}\mathit{\_}\mathit{\_}\mathit{\_}$

Short answer

Answer:

The answer is $\frac{dy}{dx}=10y$.

Step-by-step solution

Define chain rule.

The chain rule is a formula in calculus that represents the derivative of the combination of two discrete functions f and g in terms of f and g's derivatives.

Let the formula of the chain rule be, $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$. Here, $\frac{dy}{dx}$is a derivative of y with respect to x, $\frac{dy}{du}$is a derivative of y with respect to u, $\frac{du}{dx}$is a derivative of u with respect to x.

Determine the derivative of the function.

Let the derivative of the given function be, $\frac{d}{dx}{c}_1{e}^{10x}={{10}_e}^{10x}$.

Substitute $y=e^{10x}$in the above differential equation.

$\frac{dy}{dx}=10y$

Thus, the answer is $\frac{dy}{dx}=10y$.