# Problem-Solving: Differential Equations

Physics I · Differential Equations and Vector Calculus · https://tryals.app/learn/physics-i/problem-solving-differential-equations

## ODE Problem-Solving

Differential equations and vector calculus problems require choosing the right technique for each situation. The hardest part of DEs is recognising the type; once classified, the solution method is mechanical.

**ODE decision tree**
| Problem Type | Action / Technique |
| --- | --- |
| **First-order** | Check if separable, linear, exact, or Bernoulli |
| **Constant coefficients** | Write the characteristic equation |
| **Non-homogeneous** | Find $y_h$ first, then $y_p$ via undetermined coefficients |
| **Systems** | Write as $\vec{x}' = A\vec{x}$ and find eigenvalues |
| **Discontinuities** | Use Laplace transforms |

**Verification checklist:**
Substitute the solution back into the original ODE, check initial or boundary conditions, and confirm consistent units.

## Vector Calculus Decisions

Vector calculus requires picking the correct integral tool for the physical quantity you need to compute.

**Vector calculus decision tree**
| Need to compute | Tool |
| --- | --- |
| **Work along path** | Line integral $\int\vec{F}\cdot d\vec{r}$ |
| **Flux through surface** | Surface integral $\iint\vec{F}\cdot d\vec{S}$ |
| **Closed-surface flux** | Divergence theorem |
| **Circulation integral** | Stokes' theorem |
| **Conservative test** | Check if $\nabla\times\vec{F} = \vec{0}$ |

**Common pitfall:** The big theorems (Green, Stokes, divergence) trade a boundary integral for an interior one. The win comes from picking whichever side is easier. If both look equally hard, you missed a simpler choice.

## Practice questions

6 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. Complete the following passage by choosing the correct terms from the word bank.

**Answer:** The general solution of a nonhomogeneous ODE is $y = y$_h + $y_p$, where $y_h$ is the **homogeneous** solution and y_p is a **particular** solution.

**Why:** y_h solves the associated homogeneous equation; y_p satisfies the full equation.

Page: https://tryals.app/practice/physics-i/problem-solving-differential-equations/complete-the-following-passage-by-choosing-the-correct-terms-from-the

### 2. Sort each ODE by the best solution strategy.

**Answer:**

- Separation of variables: dy/dx = xy, dy/dx = y/x
- Integrating factor: dy/dx + 3$y = e^x$, dy/dx + y/$x = x^{2}$
- Characteristic equation: y'' − $y = 0$, y'' + 2y' + $y = 0$

**Why:** The equation type determines the most efficient solution method.

Page: https://tryals.app/practice/physics-i/problem-solving-differential-equations/sort-each-ode-by-the-best-solution-strategy

### 3. An RLC circuit is switched on and rings toward steady state. Order the analysis a physicist performs.

**Answer:**

1. Write Kirchhoff’s voltage law around the loop as an ODE
2. Identify it as a damped, driven second-order linear equation
3. Solve the homogeneous equation for the decaying transient
4. Find one particular solution for the steady driven response
5. Add the two and fit the initial conditions

**Why:** The transient/steady-state split is the linear-ODE worldview: the homogeneous part is the circuit’s own fading ring-down, the particular part is the response that the driver sustains forever. The same decomposition runs suspension bridges, audio filters and climate response.

Page: https://tryals.app/practice/physics-i/problem-solving-differential-equations/an-rlc-circuit-is-switched-on-and-rings-toward-steady-state-order

### 4. Choosing between separation of variables, integrating factors and Laplace transforms is usually decided before any calculation. How can the equation's form settle the method before any calculation?

A. The STRUCTURE of the equation, since each method needs a form
B. The number of initial conditions supplied
C. The physical system being modelled rather than the equation
D. Personal preference, since all three work on any equation

**Answer:** A. The STRUCTURE of the equation, since each method needs a form

**Why:** Each method has a precondition. Reading the equation’s form first is what turns method selection from guesswork into inspection.

Page: https://tryals.app/practice/physics-i/problem-solving-differential-equations/choosing-between-separation-of-variables-integrating-factors-and

### 5. The superposition principle applies to all linear differential equations, homogeneous and nonhomogeneous alike.

**Answer:** False

**Why:** False. Superposition holds for *homogeneous* linear equations. Adding two solutions of $y'' + y = f$ gives a solution of $y'' + y = 2f$, not of the original. The nonhomogeneous structure is instead 'general homogeneous solution plus one particular solution', which is a different statement.

Page: https://tryals.app/practice/physics-i/problem-solving-differential-equations/the-superposition-principle-applies-to-all-linear-differential

### 6. Which methods can solve a first-order linear ODE?

A. Integrating factor
B. Characteristic equation
C. Laplace transform
D. Separation of variables (if also separable)

**Answer:** A. Integrating factor; C. Laplace transform; D. Separation of variables (if also separable)

**Why:** The characteristic equation is for constant-coefficient equations of order $\geq 2$.

Page: https://tryals.app/practice/physics-i/problem-solving-differential-equations/which-methods-can-solve-a-first-order-linear-ode