Separation of Variables Method The general solution of a differential equation that is first-order and separable can be found using the separation of variables method.
ExercisesExercises
Separable.
Determine whether the given differential equation is separable or not. As demonstrated in the examples, if the equation is separable, use parentheses to explicitly show the separable form.
1.
\(\ds \frac{dz}{dt} = \sin(z+t) \)
Answer.Answer
not separable
2.
\(\ds s' = t\ln(s^{2t}) + 8t^2 \)
Answer.Answer
separable
3.
\(\ds \frac{dy}{dx} = 2y^3 + y + 4 \)
Answer.Answer
separable
4.
\(\ds y' - xy = 0 \)
Answer.Answer
separable
5.
\(\ds y' = x+y \)
Answer.Answer
not separable
6.
\(\ds y'' + y' + y = 0 \)
Answer.Answer
not separable
Step-by-Step.
Maybe write 1 or 2 simple scaffolded problems here?
7.
Given the differential equation \(\displaystyle \frac{dy}{dx} = xy^2 \text{,}\) determine if it is separable. If so, rewrite it in the separated form. Solution.Solution
The equation is separable because we can write it as:
\begin{equation*}
\frac{1}{y^2} dy = x dx
\end{equation*}
Answer.Answer
\(\displaystyle \frac{1}{y^2} dy = x dx \)
Continuing from the previous problem, integrate the left side of the separated equation with respect to \(y\text{.}\)Solution.Solution
Integrating \(\displaystyle \frac{1}{y^2} dy \) with respect to \(y\text{,}\) we get:
\(\displaystyle y = \frac{1}{2}\ln(4x^4 - 3) \) , or \(\displaystyle y = \ln\Big[\sqrt{4x^4 - 3}\Big] \)
Applications.
29.
A 4-lb roast, initially at \(50^{\circ}F\text{,}\) is placed in a \(375^{\circ} F\) oven at 5:00 PM After 75 minutes it is found that the temperature \(T(t)\) of the roast is \(125^{\circ}\) F. When will the roast be \(150^{\circ}F\) (medium rate)?
Answer.Answer
\(t = -[\ln(225/325)]/[0.0035] \approx 105(min)\)
Conceptual Questions.
30.
Explain why separating variables works as a method for solving ordinary differential equations.
31.
Why is it important for the equation to be "separable" in order to use the method of separation of variables? What does it mean for a differential equation to be separable?
32.
Provide an example of a differential equation that is not separable and explain why the method of separation of variables cannot be applied to it.
33.
In solving a differential equation using separation of variables, why might the constant of integration appear on both sides of the resulting equation?
34.
Some differential equations solved using separation of variables yield implicit solutions, while others yield explicit solutions. Explain the difference between implicit and explicit solutions, and provide examples of each.
35.
True/False: Every first-order ordinary differential equation can be solved using the method of separation of variables.
36.
Which of the following differential equations is not separable?