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Appendix F Selected Hints
1 What is a Differential Equation? 1.5 Exercises 🏋️♂️ Practice Drills
Exercises
1. Identify the Differential Equations.
1.a Hint .
There are only four differential equations in this set.
3. Identify the Coefficients.
3.b Hint .
Look for the dependent variable in each term. The coefficient is the constant or function that multiplies the dependent variable.
2 Classification 2.4 Exercises 🏋️♂️ Practice Drills
Exercises
1. Identify the Linear & Nonlinear Terms.
1.e
2. Identify the Linear & Nonlinear Differential Equations.
2.f Hint .
Remember that a linear differential equation contains only linear terms. Four of these equations are linear.
2.g Hint .
First, identify the dependent variable, then carefully examine each term to determine whether it is nonlinear.
5 Separation of Variables 5.5 Exercises 💡 Conceptual Quiz
Exercises
1. True or False.
1.i 👍👎.
Hint .
Check that the differential equation is first-order and separable.
1.j 👍👎.
Hint .
Check that the differential equation is first-order and separable.
6 Integrating Factor 6.1 Exercises
💡 Conceptual Quiz 6.1.1.
6.1.1.a 👍👎.
Hint .
Check to see that the differential equation is in the form
\(y'+P(x)y=Q(x)\text{.}\)
🏋️♂️ Practice Drills
6.1.1. Identifying Equations for the IF Method.
Hint .
A first-order linear equation has
\(y\) and
\(y'\) only to the first power and not inside nonlinear functions.
8 Euler’s Method 8.4 Exercises
✍🏻 Problems 8.4.1. Hint .
Start with the given initial condition
\(y_0 = -1\) at
\(x_0 = 0\text{.}\) Use Euler’s update formula:
\(y_{k+1} = y_k + h \cdot f(x_k, y_k)\text{,}\) where
\(f(x, y) = y^2 - x\text{.}\) You’ll need to compute two steps to reach
\(x = 1\text{.}\)
9 Homogeneous Equations (LHCC) 9.6 Exercises
🏋️♂️ Practice Drills
9.6.1. Identifying LHCC Equations.
9.6.1.f Select the LHCC Equations.
Hint .
There are only 4 LHCC equations in this set.
