Differential Equations Quiz: Technology For Solutions

What this quiz covers

This quiz focuses on Technology For Solutions, giving you a quick way to practice the rules, question types, and explanations that matter most for Differential Equations.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. First-order ($O(h)$)
2. Second-order ($O(h^2)$)
3. Third-order ($O(h^3)$)
4. Fourth-order ($O(h^4)$) (correct answer)

Explanation: The global error $E$ for a method of order $p$ is proportional to $h^p$, so $E \approx C h^p$. We can write the ratio of the errors as $\frac{E_2}{E_1} \approx \frac{C (h_2)^p}{C (h_1)^p} = (\frac{h_2}{h_1})^p$. Here, $h_1 = 0.1$ and $h_2 = 0.05$, so $\frac{h_2}{h_1} = \frac{0.05}{0.1} = \frac{1}{2}$. The ratio of the errors is $\frac{1.60 \times 10^{-3}}{2.56 \times 10^{-2}} = \frac{0.0016}{0.0256} = \frac{1}{16}$. So, we have $\frac{1}{16} \approx (\frac{1}{2})^p$. This implies $2^p \approx 16$, which means $p=4$. Therefore, the method is fourth-order, such as the classic Runge-Kutta method.

Question 2

1. A spiral that moves outward, away from the origin. (correct answer)
2. A trajectory that converges to the equilibrium point at the origin.
3. A closed loop that closely approximates the true circular path.
4. A spiral that moves inward, toward the origin.

Explanation: When analyzing numerical methods for differential equations, you need to understand how computational errors accumulate over time, especially for systems that should conserve energy. The given system $x' = -y, y' = x$ represents simple harmonic motion in two dimensions. The true solution traces a perfect circle of radius 2, meaning the system conserves energy: $x^2 + y^2 = 4$ for all time. However, the forward Euler method introduces systematic errors. At each step, Euler's method uses $x_{n+1} = x_n + h(-y_n)$ and $y_{n+1} = y_n + h(x_n)$. This creates a slight increase in the distance from the origin at each time step. You can verify this by computing $x_{n+1}^2 + y_{n+1}^2 = x_n^2 + y_n^2 + h^2(x_n^2 + y_n^2)$, showing the "energy" grows by factor $(1 + h^2)$ each step. Over many steps, this creates an outward spiral. Choice A is correct because the forward Euler method's truncation error causes the trajectory to spiral outward from the true circular path. Choice B is wrong because the method doesn't create damping—it actually adds energy to the system. Choice C is incorrect because while the trajectory might look circular initially, the systematic energy increase prevents it from remaining a closed loop. Choice D represents what would happen with backward Euler, which tends to be energy-dissipating rather than energy-increasing. Study tip: Remember that forward Euler tends to be unstable for oscillatory systems, typically adding energy and causing outward spirals, while implicit methods like backward Euler tend to dissipate energy.

Question 3

1. -0.538
2. 0.000
3. 0.538 (correct answer)
4. 1.000

Explanation: This question requires recognizing the behavior of the solution or performing a quick approximation with a simple method like Euler's. Let's use Euler's method with a reasonably small step size, say $h=0.5$, to get an estimate. The IVP is $y' = 1 - 2xy$, $y(0)=0$. Step 1: $(x_0, y_0) = (0,0)$. The slope is $y'(0) = 1 - 2(0)(0) = 1$. The next point is $y(0.5) \approx y_0 + h \cdot y'(0) = 0 + 0.5(1) = 0.5$. Step 2: $(x_1, y_1) = (0.5, 0.5)$. The slope is $y'(0.5) = 1 - 2(0.5)(0.5) = 1 - 0.5 = 0.5$. The next point is $y(1.0) \approx y_1 + h \cdot y'(0.5) = 0.5 + 0.5(0.5) = 0.75$. This crude approximation (0.75) is positive and closer to 0.538 than the other options. The true solution is related to the error function, and $y(1) \approx 0.538$. Let's analyze the distractors. 0.000 would imply the function doesn't change from its initial value. 1.000 would happen if the slope remained 1 for the whole interval. -0.538 is a sign error. The initial slope is positive, and for small $x, y$, the slope remains positive, so the function must increase initially. The value 0.538 is the only plausible positive result.

Question 4

1. $|y_h(5) - y_{h/2}(5)|$
2. $\frac{1}{15}|y_h(5) - y_{h/2}(5)|$ (correct answer)
3. $\frac{1}{16}|y_h(5) - y_{h/2}(5)|$
4. $\frac{1}{2}|y_h(5) - y_{h/2}(5)|$

Explanation: Let $Y(x)$ be the true solution. For a fourth-order method, the global error is approximately $E(x) \approx C h^4$. So, $Y(5) \approx y_h(5) + C h^4$ and $Y(5) \approx y_{h/2}(5) + C (h/2)^4 = y_{h/2}(5) + C h^4/16$. Let $E_{h/2} = Y(5) - y_{h/2}(5) \approx C h^4/16$. We want to find an expression for this error. Subtracting the two approximate equations gives $0 \approx (y_h(5) - y_{h/2}(5)) + C h^4 (1 - 1/16) = (y_h(5) - y_{h/2}(5)) + (15/16) C h^4$. This means $C h^4 \approx -\frac{16}{15}(y_h(5) - y_{h/2}(5))$. Substituting this back into the expression for the error $E_{h/2}$: $E_{h/2} \approx \frac{1}{16} C h^4 \approx \frac{1}{16} [-\frac{16}{15}(y_h(5) - y_{h/2}(5))] = -\frac{1}{15}(y_h(5) - y_{h/2}(5))$. The magnitude of the error is therefore $\frac{1}{15}|y_h(5) - y_{h/2}(5)|$. This technique is known as Richardson extrapolation.

Question 5

1. Underestimate, because the slope $y'$ is positive at the initial point.
2. Overestimate, because the slope $y'$ is positive at the initial point.
3. Underestimate, because the solution curve is concave up near the initial point.
4. Overestimate, because the solution curve is concave down near the initial point. (correct answer)

Explanation: Euler's method approximates the solution by following the tangent line. Whether it overestimates or underestimates depends on the concavity of the solution curve. We can find the concavity by computing the second derivative. Given $y' = x - y^2$, we differentiate with respect to $x$: $y'' = \frac{d}{dx}(x - y^2) = 1 - 2y \cdot y'$. At the initial point $(x_0, y_0) = (2, 1)$, we have $y'(2) = 2 - 1^2 = 1$. Now, we evaluate $y''$ at this point: $y''(2) = 1 - 2(1)(1) = -1$. Since $y'' < 0$, the solution curve is concave down near $(2, 1)$. For a concave down curve, the tangent line lies above the curve, so Euler's method will produce an overestimation.

Question 6

1. The global truncation error is a complex accumulation of local truncation errors, where errors from earlier steps can be amplified or diminished by the differential equation. (correct answer)
2. The global truncation error is the direct sum of all local truncation errors from previous steps, with no interaction between errors.
3. The global truncation error equals the local truncation error from the final step only, with no contribution from previous steps.
4. The global and local truncation errors are independent quantities determined solely by step size and method order respectively, with no relationship.

Explanation: When analyzing numerical methods for differential equations, understanding how errors propagate is crucial for assessing solution quality. Both local and global truncation errors matter, but their relationship is more complex than simple addition. The correct answer is A because global truncation error involves a sophisticated accumulation process. Each local error from previous steps doesn't just add to the total—it gets carried forward and can be amplified or dampened by subsequent calculations. The differential equation itself influences how these errors evolve. For example, if $f(x,y)$ has a large derivative with respect to $y$, small errors in earlier $y$-values can grow exponentially. Conversely, some differential equations naturally damp errors over time. Choice B incorrectly suggests simple linear accumulation. This ignores how the iterative nature of numerical methods means that errors from step $n$ affect the starting point for step $n+1$, creating a compounding effect rather than mere addition. Choice C is fundamentally wrong because it dismisses all previous computational history. Global error inherently reflects the cumulative effect of the entire solution process. Choice D treats these errors as completely separate entities, which misses the core relationship: global error emerges precisely from how local errors interact through the stepping process. Study tip: Remember that numerical methods are inherently iterative—each step builds on previous results. This means errors have a "memory" that propagates forward, making global error analysis much more nuanced than simply counting individual step errors.

Question 7

1. The step size is too large for this particular differential equation, causing numerical instability in regions where the derivative changes rapidly (correct answer)
2. Euler's method cannot be applied to nonlinear differential equations like this one, so a different numerical method must be used
3. The initial condition $y(0) = 0.5$ is incompatible with the differential equation $\frac{dy}{dt} = t^2 - y^2$
4. The differential equation has no solution that passes through the point $(0, 0.5)$, so numerical methods will always fail

Explanation: Euler's method can exhibit poor accuracy or apparent divergence when the step size is too large, especially for equations where the derivative changes rapidly. The equation $\frac{dy}{dt} = t^2 - y^2$ is nonlinear and can have regions where the derivative changes quickly, making a smaller step size necessary for accurate approximation. Option B is incorrect because Euler's method works for nonlinear equations. Option C is wrong because initial conditions are typically compatible with differential equations unless there's a specific constraint. Option D is incorrect because the existence and uniqueness theorem guarantees a solution exists for this smooth function.

Question 8

1. The spirals will maintain the same direction of rotation but will spiral inward toward the origin instead of outward
2. The spirals will reverse their direction of rotation while continuing to spiral inward toward the origin as before
3. The spirals will maintain the same direction of rotation but will spiral outward away from the origin instead of inward (correct answer)
4. The trajectories will change from spirals to straight lines radiating outward from the origin in all directions

Explanation: In the original system, the $-0.1y$ term provides damping, causing inward spiraling. When changed to $+0.1y$, this becomes anti-damping, causing outward spiraling. The direction of rotation remains the same because the primary terms $\frac{dx}{dt} = y$ and $\frac{dy}{dt} = -x$ are unchanged. Option A reverses the inward/outward behavior. Option B incorrectly suggests the rotation direction changes. Option D is incorrect because the system still has oscillatory behavior, just with growing amplitude.

Question 9

1. The adaptive algorithm is malfunctioning because the step size should increase near points where the solution becomes large
2. The step size reduction indicates that the solution is becoming more linear, requiring finer discretization for accurate approximation
3. The algorithm detects oscillatory behavior in the solution and reduces step size to resolve the high-frequency components
4. The solution is approaching a vertical asymptote, and the algorithm reduces step size to maintain accuracy as $\frac{dy}{dx}$ becomes very large (correct answer)

Explanation: When you encounter questions about adaptive step-size control in numerical methods, think about why algorithms automatically adjust their behavior. These methods monitor local truncation error and modify step sizes to maintain accuracy while managing computational efficiency. The differential equation $\frac{dy}{dx} = x^2 + y^2$ with $y(0) = 0$ is notorious for developing finite-time blowup. As the solution grows, $y^2$ becomes increasingly large, causing $\frac{dy}{dx}$ to grow explosively. Near $x = 1.4$, the solution is approaching a vertical asymptote where the derivative becomes infinite. The adaptive algorithm detects this rapid change by monitoring the local error estimates, which spike as the solution steepens dramatically. To maintain accuracy in this challenging region, the algorithm automatically reduces step size, trying to track the solution's sharp ascent toward infinity. Option A is backwards—step sizes should decrease, not increase, when solutions become large and change rapidly. Option B mischaracterizes the behavior; the solution is becoming highly nonlinear near blowup, not more linear. A more linear solution would actually allow larger step sizes. Option C incorrectly identifies oscillations as the cause. This equation doesn't produce oscillatory solutions; instead, it creates monotonically increasing solutions that blow up. For differential equations exams, remember that adaptive algorithms are your diagnostic tools. When step sizes decrease dramatically, ask yourself: "What's making the solution hard to approximate?" Usually it's either rapid change (like approaching asymptotes) or high curvature—both signal that the solution is becoming numerically challenging.

Question 10

1. The equilibrium is a stable spiral since one eigenvalue is positive and one is negative, creating rotational motion that eventually converges
2. The linearization is inconclusive since the eigenvalues are too close to zero, requiring higher-order analysis to determine stability
3. The equilibrium is an unstable node since both eigenvalues are real and at least one is positive, causing all nearby trajectories to diverge
4. The equilibrium is a saddle point since the eigenvalues have opposite signs, making it unstable with trajectories approaching along one direction and departing along another (correct answer)

Explanation: When analyzing equilibrium points in nonlinear systems, the eigenvalues of the Jacobian matrix at each equilibrium determine the local stability and behavior type. The key insight is that real eigenvalues with opposite signs create a very specific geometric pattern. With eigenvalues $\lambda_1 = 1$ and $\lambda_2 = -0.5$, you have one positive and one negative real eigenvalue. This configuration creates a saddle point - an inherently unstable equilibrium where trajectories approach along the direction of the negative eigenvalue (the stable manifold) but diverge along the direction of the positive eigenvalue (the unstable manifold). Picture a mountain pass where paths converge from one direction but diverge in the perpendicular direction. Answer A incorrectly suggests spiral behavior, but spirals require complex eigenvalues with nonzero imaginary parts. Real eigenvalues like these produce straight-line motion along eigenvector directions, not rotational motion. Answer B misunderstands the significance of these eigenvalues. Neither value is close to zero, so the linearization gives definitive information about local behavior. Values near zero would indeed make linearization inconclusive. Answer C correctly identifies instability but wrongly classifies this as a node. Nodes occur when eigenvalues have the same sign. Here, opposite signs definitively indicate a saddle point, not a node. Remember: eigenvalue signs determine stability classification. Same signs = node (stable if both negative, unstable if both positive), opposite signs = saddle (always unstable), complex with nonzero imaginary parts = spiral or focus.

Question 11

1. Solutions cross these curves smoothly without any change in behavior since the direction field is only temporarily undefined
2. These curves represent isoclines where all direction vectors have the same finite slope value throughout the region
3. Solutions approach these curves asymptotically but cannot cross them, as the differential equation becomes singular along these lines (correct answer)
4. These curves indicate regions where the computer algebra system has computational errors and should be ignored in the analysis

Explanation: The direction field has undefined slopes where the denominator $x - y = 0$, or along the line $y = x$. At these points, the differential equation becomes singular (undefined), and solutions typically approach such lines asymptotically but cannot cross them. Option A is incorrect because solutions cannot simply cross singular lines. Option B is wrong because these are not isoclines (lines of constant slope) but rather lines where the slope is undefined. Option D is incorrect because this is a mathematical property of the differential equation, not a computational error.

Question 12

1. The shooting method is inappropriate for this type of linear boundary value problem and a different numerical approach must be used
2. The problem has infinitely many solutions because the boundary conditions are satisfied by $y = B\sin(x)$ for any constant $B$ (correct answer)
3. The computational software has numerical precision errors that prevent convergence to the unique solution $y = 0$
4. The boundary conditions are inconsistent with the differential equation, making the problem mathematically impossible to solve

Explanation: Applying the boundary conditions: $y(0) = A\cos(0) + B\sin(0) = A = 0$, so $y = B\sin(x)$. Then $y(\pi) = B\sin(\pi) = 0$, which is satisfied for any value of $B$. This means there are infinitely many solutions, explaining why the shooting method cannot converge to a unique solution. Option A is wrong because the shooting method works for linear BVPs when solutions are unique. Option C is incorrect because this is a mathematical property, not a numerical precision issue. Option D is wrong because the conditions are consistent, just not sufficient to determine a unique solution.

Question 13

1. Approximately $0.0001$, since halving the step size reduces the error by half
2. Approximately $0.0000125$, since the fourth-order method means errors scale as $h^4$ (correct answer)
3. Approximately $0.00004$, since the error reduction follows a quadratic relationship with step size
4. Approximately $0.000004$, since each halving of step size reduces error by a factor of about 50

Explanation: The fourth-order Runge-Kutta method has a global error that scales as $h^4$. When the step size is halved, the error is reduced by a factor of $2^4 = 16$. Since the difference between $h = 0.1$ and $h = 0.05$ is $0.0002$, when going from $h = 0.05$ to $h = 0.025$, we expect the difference to be approximately $0.0002/16 ≈ 0.0000125$. Option A uses first-order scaling, Option C uses second-order scaling, and Option D uses an incorrect scaling factor.

Question 14

1. Both solutions show polynomial-like behavior near the origin with different limiting values (correct answer)
2. One solution remains finite while the other has a logarithmic singularity at the origin
3. Both solutions exhibit oscillatory behavior with increasing frequency approaching zero
4. One solution diverges like $x^{-1/2}$ while the other remains bounded near zero

Explanation: For half-integer order Bessel functions, $J_{1/2}(x) \sim \sqrt{\frac{2}{\pi x}}\sin(x)$ and $J_{-1/2}(x) \sim \sqrt{\frac{2}{\pi x}}\cos(x)$. Near $x = 0$, these behave like $\sqrt{x}$ and $x^{-1/2}$ respectively, but when expressed as power series solutions, they appear polynomial-like in their local behavior. Option B describes logarithmic singularities that don't occur here. Option C incorrectly describes the oscillatory behavior. Option D overstates the singular behavior visible in typical plotting ranges.