Q8E Solve the nonlinear differential... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q8E (page 425)

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{d x}{d t} = f x$ as $\frac{d x}{f x} = d t$ and integrate both sides.
8. $\frac{d x}{d t} = \sqrt{x}, x (0) = 4$

Short Answer

Expert verified

The solution is $x (t) = \frac{t^{2}}{4} + 4 + 2 t$ .

Step by step solution

Simplification for the differential equation

Consider the equation as follows.

$\frac{d x}{d t} = \sqrt{x}$

Now, separate the variables as follows.

$\begin{matrix} \frac{d x}{d t} = \sqrt{x} \\ \frac{d x}{\sqrt{x}} 鈥� = d t \\ x^{\frac{- 1}{2}} d x = d t \end{matrix}$

Integrating on both sides as follows.

$\begin{matrix} x^{\frac{- 1}{2}} d x = d t \\ 鈭� x^{\frac{- 1}{2}} d x = 鈭� d t \\ \frac{x^{\frac{- 1}{2} + 1}}{\frac{- 1}{2} + 1} = t + C \\ \frac{x^{\frac{- 1 + 2}{2}}}{\frac{- 1 + 2}{2}} = t + C \end{matrix}$

Simplify further as follows:

$\begin{matrix} \frac{x^{\frac{- 1 + 2}{2}}}{\frac{- 1 + 2}{2}} = t + C \\ \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = t + C \\ 2 x^{\frac{1}{2}} = t + C \end{matrix}$

Substituting the initial condition as follows.

$\begin{matrix} 2 x^{\frac{1}{2}} = t + C \\ 2 {(4)}^{\frac{1}{2}} = 0 + C 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� {鈭� x (0) = 4} \\ 2 (2) = C \\ 4 = C \end{matrix}$

Calculation of the solution

Now, substitute the value $4$ for $c$ in $2 x^{\frac{1}{2}} = t + C$ as follows:

$\begin{array}{l} 2 x^{\frac{1}{2}} = t + C \\ 2 x^{\frac{1}{2}} = t + 4 \\ x^{\frac{1}{2}} = \frac{1}{2} (t + 4) \\ x^{\frac{1}{2}} = \frac{t}{2} + \frac{4}{2} \end{array}$

Simplify further as follows.

$\begin{array}{l} x^{\frac{1}{2}} = \frac{t}{2} + \frac{4}{2} \\ x^{\frac{1}{2}} = \frac{t}{2} + 2 \end{array}$

Now, squaring on both sides as follows:

$\begin{matrix} x^{\frac{1}{2}} = \frac{t}{2} + 2 \\ {(x^{\frac{1}{2}})}^{2} = {(\frac{t}{2} + 2)}^{2} \\ x = \frac{t^{2}}{4} + 4 + 2 脳 2 脳 \frac{t}{2} \\ x (t) = \frac{t^{2}}{4} + 4 + 2 t \end{matrix}$

Hence, the solution for the differential equation $\frac{d x}{d t} = \sqrt{x}$ is $x (t) = \frac{t^{2}}{4} + 4 + 2 t$ .

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91影视

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{d x}{d t} = f x$ as $\frac{d x}{f x} = d t$ and integrate both sides.
8. $\frac{d x}{d t} = \sqrt{x}, x (0) = 4$

Short Answer

Step by step solution

Simplification for the differential equation

Calculation of the solution

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91影视

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation dxdt=fxas dxfx=dtand integrate both sides.8.dxdt=x,x(0)=4

Short Answer

Step by step solution

Simplification for the differential equation

Calculation of the solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Applied Mathematics

Mechanics Maths

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{d x}{d t} = f x$ as $\frac{d x}{f x} = d t$ and integrate both sides.
8. $\frac{d x}{d t} = \sqrt{x}, x (0) = 4$