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Section Chapter 4 Exercises
Reading Questions แฏโ
โ Quick-Answer Questions
1. True-False.
(a) ๐๐.
The differential equation, below, is in standard form.
\begin{equation*}
xy'+2xy=x^2
\end{equation*}
True.
This differential equation is not in the form \(y'+P(x)y=Q(x)\text{.}\)
False.
This differential equation is not in the form \(y'+P(x)y=Q(x)\text{.}\)
Hint .
Check to see that the differential equation is in the form
\(y'+P(x)y=Q(x)\text{.}\)
2. Multiple-Choice.
(a) Integration Technique Used.
What integration technique is used to evaluate
\begin{equation*}
\int t^3 \ln t \, dt\text{?}
\end{equation*}
Substitution.
Substitution isnโt the best choice for this integral.
Integration by parts.
Correct! Integration by parts handles products like \(t^3 \ln t\text{.}\)
Partial fraction decomposition.
Partial fractions donโt apply here.
The product rule.
The product rule is for differentiation, not integration.
(b) When to Use Direct Integration.
What method is used to solve a differential equation of the form
\begin{equation*}
\frac{d}{dx}[g(x,y)] = f(x)\text{?}
\end{equation*}
Direct integration.
Correct! Integrate both sides with respect to \(x\text{.}\)
Separation of variables.
Separation isnโt neededโthe equation integrates directly.
Substitution.
Substitution isnโt the primary method here.
Partial fractions.
Partial fractions are sometimes used inside integration, but the method here is simply direct integration.
(c) Conditions for Using the Integrating Factor Method.
What properties must a differential equation have for the integrating factor method to apply?
First order.
Correct! IF method is for first-order equations.
Separable.
Separable equations donโt require the integrating factor method.
Linear.
Correct! The IF method works only for linear equations.
Quadratic.
โQuadraticโ refers to algebraic equations, not differential equations.
(d) Choosing the Integrating Factor.
For the linear equation
\(y' + 3y = x\text{,}\) what is the integrating factor?
\(\ds e^{\int 3 \, dx} = e^{3x}\) Correct! The integrating factor is built from \(P(x) = 3\text{.}\)
\(\ds e^{\int x \, dx} = e^{x^2/2}\) This would come from \(Q(x)\text{,}\) not \(P(x)\) โwrong choice.
\(\ds e^{-3x}\) Watch the signโthe integrating factor has \(e^{+\int P(x) dx}\text{.}\)
\(\ds 3e^x\) The integrating factor isnโt just an arbitrary exponentialโit comes from integrating \(P(x)\text{.}\)
(e) Purpose of Multiplying by the Integrating Factor.
Why do we multiply a first-order linear equation by its integrating factor?
To rewrite the equation so it can be integrated directly.
Correct! The integrating factor allows the left-hand side to become \(\frac{d}{dx}[\mu(x)y]\text{,}\) which is integrable.
To convert the equation into a separable equation.
Not quiteโalthough integration becomes possible, the equation doesnโt become separable.
To set up the product rule for solving the equation.
The product rule is reversed here, not applied to expand further.
To make the equation factorable.
The integrating factor rewrites the equation into an integrable form, not just a โfactorableโ one.
(f) Next Step After Using the Integrating Factor.
You rewrote \(\dfrac{dy}{dx} + 2y = 5\) as
\begin{equation*}
\frac{d}{dx} \left[e^{2x} y\right] = 5e^{2x}\text{.}
\end{equation*}
What should you do next?
Apply the product rule to expand the left-hand side.
The product rule was already reversed to reach this form; expanding again would undo progress.
Integrate both sides to solve for \(y\text{.}\)
Correct! Once the left-hand side is a single derivative, integration gives the solution for \(y\text{.}\)
Multiply by another integrating factor.
Only one integrating factor is needed. Multiplying by another one is unnecessary.
Differentiate both sides to simplify the expression.
Further differentiation is not needed; integration is the next step.
(g) Choosing the IF for a Given Equation.
Find the integrating factor for
\begin{equation*}
x^2 y' - y = 1\text{.}
\end{equation*}
\(e^{1/x^2}\) Not quiteโcheck the exponent carefully.
\(x^2\) This is not the exponential integrating factor needed.
\(e^{1/x}\) Correct! The integrating factor comes from \(P(x) = -\frac{1}{x^2}\text{.}\)
\(\frac{1}{x}\) This is not the correct integrating factor.
(h) Sequence of IF Method Steps.
Which of the following best represents a succinct version of the integrating factor method? Let
\(\mu\) be the integrating factor.
standard formโcompute \(\mu\) โmultiply \(\mu\) โintegrateโisolate
Correct! Thatโs the sequence every time.
compute \(\mu\) โmultiply \(\mu\) โdifferentiateโisolate
Differentiation is not used.
integrateโcompute \(\mu\) โmultiply \(\mu\) โstandard formโisolate.
Correct steps, wrong order.
isolateโstandard formโcompute \(\mu\) โmultiply \(\mu\) โintegrate.
Correct steps, wrong order.
3. Short-Answer.
(a) Classification of DEs for the Integrating Factor Method.
What type of differential equations can be solved using the integrating factor method?
(b) Equations Outside Separation and IF Methods.
Classify all first-order differential equations that cannot be solved by either separation of variables or the integrating factor method. Provide an example of such an equation.
(c) Rewrite into Standard Linear Form.
4. Other.
(a) Identifying Equations for the IF Method.
Which of the following differential equations can be solved using the integrating factor method?
Four of these equations are linear and first-orderโthey can be solved by the integrating factor method.
\(y' + 2y = 3x\) \(\dfrac{y'}{t^2} + y = 17t\) \(y'' + 2x y = 0\)
\(\)
\(y' + y^2 = 17t\) \(\dfrac{y'}{t} + y = 17t\) \(\cos x \, y = y' - e^x\)
Hint .
A first-order linear equation has
\(y\) and
\(y'\) only to the first power and not inside nonlinear functions.
Exercises ๐๏ธ Warm-ups & Drills
Which Method Applies.
For each differential equation below, identify which statement applies and explain how you know.
You do not need to solve any of the equations.
Can be solved by separating variables, but not using an integrating factor.
Can be solved using an integrating factor, but not by separating variables.
Can be solved both by using an integrating factor and by separating variables.
Cannot be solved using either technique.
1. \(x^2 \dfrac{dy}{dx} + \cos x = y\) 2. \(\dfrac{dx}{dt} + xt = e^x\) 3. \((t^2 + 1) \dfrac{dy}{dt} = yt - y\) 4. \(\dfrac{dy}{dt} - y^2 t = t\) 5. \(3r = \dfrac{dr}{d\theta} - \theta^3\)
Exercises โ๐ป Solve the Differential Equations
General Solution. Use an integrating factor to find the general solution to each equation.
1. \(y' - 4y = x\) 2. \(x^2 y' - y = 1, \quad x > 0\) 3. \(\dfrac{dy}{dx} - y = e^{3x}\) 4. \(\dfrac{dy}{dx} = \dfrac{y}{x} + 2x + 1,\quad x > 0\) 5. \(y\dfrac{dx}{dy} + 2x = 5y^3 \) 6. \(-\dfrac{4M}{t^2} = t^5 - \dfrac{1}{t}\frac{dM}{dt} + 1\) 7. \(\dfrac{dy}{dx} + 4x^3 y = 5xe^{-x^4}\) 8. \(\dfrac{dr}{d\theta} + r\tan \theta = \sec \theta,\ \ -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}\)
Initial-Value Problems. Solve each initial value problem.
9. \(\dfrac{dy}{dx} - \dfrac{y}{x} = xe^x,\quad y(1) = e - 1 \quad (x > 1)\) 10. \(e^t z' = 1 - 4e^t z,\quad z(0) = \dfrac{4}{3}\) 11. \(y' - 3y = 6,\quad y(0) = 5\)
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