Question

Answer Problems 1-12 without referring back to the text. Fill in the blanks or answer true or false.

The initial-value problem $\mathit{x}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{-}\mathbf{4}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathit{y}\left(0\right)\mathbf{=}\mathit{k}$, has an infinite number of solutions for k =and no solution fork = .

Short answer

Infinite solution for k = 0 & no solution for $k\ne 0$.

Step-by-step solution

Definition

A differential equation is an equation with one or more derivatives of a function.

Solve equation

Separate variables to get$\frac{dy}{y}=\frac{4}{x}dx$

$\begin{array}{rcl}I{n}^{\left|y\right|}& =& 4In\left|x\right|+c\\ & =& I{n}^{x^4}+c\\ & =& I{n}^{c_4{x}^4}\\ y& =& c_1{x}^4\end{array}$

The condition y(0) Implies k = 0

Thus, the initial-value problem has an infinite number of solutions for k = 0 and no solutions for $k\ne 0$.