1.
(a) 👍👎.
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 \(y'+P(x)y=Q(x)\text{.}\)
The differential equation, below, is in standard form.
\begin{equation*}
xy'+2xy=x^2
\end{equation*}
(b) 👍👎.
True.
- Since the term, \(y^2\text{,}\) is nonlinear, the differential equation is nonlinear and the integrating factor method does not apply here.
False.
- Since the term, \(y^2\text{,}\) is nonlinear, the differential equation is nonlinear and the integrating factor method does not apply here.
The integrating factor method can be applied to the differential equation
\begin{equation*}
y' = y^2 + x\text{.}
\end{equation*}
(c) Integration Technique Used.
What integration technique is used to evaluate
\begin{equation*}
\int t^3 \ln t \, dt\text{?}
\end{equation*}
- Substitution.
- Substitution isn’t 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 don’t apply here.
- The product rule.
- The product rule is for differentiation, not integration.
(d) 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 isn’t needed—the equation integrates directly.
- Substitution.
- Substitution isn’t the primary method here.
- Partial fractions.
- Partial fractions are sometimes used inside integration, but the method here is simply direct integration.
(e) 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 don’t require the integrating factor method.
- Linear.
- Correct! The IF method works only for linear equations.
- Quadratic.
- “Quadratic” refers to algebraic equations, not differential equations.
(f) 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 \(P(x)\)—wrong choice.
- \(\ds e^{-3x}\)
- Watch the sign—the integrating factor has \(e^{+\int P(x) dx}\text{.}\)
- \(\ds 3e^x\)
- The integrating factor isn’t just an arbitrary exponential—it comes from integrating \(P(x)\text{.}\)
(g) 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 quite—although integration becomes possible, the equation doesn’t 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 “factorable” one.
(h) 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.
(i) Choosing the IF for a Given Equation.
- \(e^{1/x^2}\)
- Not quite—check 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.
(j) 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 form—compute \(\mu\)—multiply \(\mu\)—integrate—isolate
- Correct! That’s the sequence every time.
- compute \(\mu\)—multiply \(\mu\)—differentiate—isolate
- Differentiation is not used.
- integrate—compute \(\mu\)—multiply \(\mu\)—standard form—isolate.
- Correct steps, wrong order.
- isolate—standard form—compute \(\mu\)—multiply \(\mu\)—integrate.
- Correct steps, wrong order.
(k) Classification of DEs for the Integrating Factor Method.
What type of differential equations can be solved using the integrating factor method?
(l) 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.
