Chapter 2: Q2.3-12E (page 54)

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dy}{dx} = x^{2} e^{- 4 x} - 4 y$

Short Answer

Expert verified

The general solution to the given equation is $y = \frac{x^{3}}{3} e^{- 4 x} + {Ce}^{- 4 x}$

Step by step solution

Method for solving linear equations

Write the equation in the standard form $\frac{dy}{dx} + P (x) y = Q (x)$ .
Calculate the integrating factor by $μ (x)$ the formula .
Multiply the equation in standard form by $μ (x)$ and, recalling that the left-hand side is just $\frac{d}{dx} [μ (x) y]$ , obtain

$μ (x) \frac{dy}{dx} + P μ (x) y = μ (x) Q (x) \frac{d}{dx} [μ (x) y] = μ (x) Q (x)$

Integrate the last equation and solve for y by dividing by $μ (x)$ to obtain $y (x) = \frac{1}{π (x)} [\int_{}^{} π (x) Q (x) + c]$ . Here C is an arbitrary constant.

Step 2: Solve the given equation

Given that,

$\frac{dy}{dx} = x^{2} e^{- 4 x} - 4 y . . . . . . . (1)$

Write the equation (1) into standard form of linear equation.

$\frac{dy}{dx} = x^{2} e^{- 4 x} - 4 y \frac{dy}{dx} + 4 y = x^{2} e^{- 4 x} . . . . . . . . . . (2)$

Calculate the integrating factor of $μ (x)$ .

Where $P (x) = 4$ .

Then,

$\begin{array}{rcl} μ (x) & = & e^{\int_{}^{} P (x) dx} \\ = & e^{\int_{}^{} 4 dx} \\ = & e^{4 x} \end{array}$

Step 3: Simplification method

Multiply $μ (x)$ in equation (2)

$\begin{array}{rcl} e^{4 x} \frac{dy}{dx} + 4 e^{4 x} y & = & x^{2} e^{4 x} e^{- 4 x} \\ e^{4 x} \frac{dy}{dx} + 4 e^{4 x} y & = & x^{2} \\ \frac{d}{dx} [e^{4 x} y] & = & x^{2} \end{array}$

Integrating both sides.

role="math" localid="1664260665464" $\begin{array}{rcl} \int_{}^{} \frac{d}{dx} [e^{4 x} y] d x & = & \int_{}^{} x^{2} dx \\ e^{4 x} y & = & \frac{x^{3}}{3} + C \\ y & = & \frac{x^{3}}{3} e^{- 4 x} + {Ce}^{- 4 x} \end{array}$

So, the solution is $\begin{array}{rcl} y & = & \frac{x^{3}}{3} e^{- 4 x} + {Ce}^{- 4 x} \end{array}$ .

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Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dy}{dx} = x^{2} e^{- 4 x} - 4 y$

Short Answer

Step by step solution

Method for solving linear equations

Step 2: Solve the given equation

Step 3: Simplification method

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Question: In Problems 7-16, obtain the general solution to the equation.dydx=x2e-4x-4y

Short Answer

Step by step solution

Method for solving linear equations

Step 2: Solve the given equation

Step 3: Simplification method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Mechanics Maths

Discrete Mathematics

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dy}{dx} = x^{2} e^{- 4 x} - 4 y$