Differential Equations Quiz: Slope Fields

What this quiz covers

This quiz focuses on Slope Fields, 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. $dy/dx = x/y$
2. $dy/dx = y/x$ (correct answer)
3. $dy/dx = xy$
4. $dy/dx = x^2 - y^2$

Explanation: Let's translate the given properties into mathematical conditions on $f(x,y)$.

1. 'Slopes are zero on the x-axis' means $f(x,0)=0$ for $x \neq 0$.
2. 'Slopes are undefined on the y-axis' means $f(0,y)$ is undefined for $y \neq 0$.
3. 'Slopes are constant on any line through the origin' means the value of $f(x,y)$ depends only on the ratio $y/x$. Such a function is called a homogeneous function of degree 0. Let's test the options: A) $f(x,y)=x/y$. $f(x,0)$ is undefined (fails 1). $f(0,y)=0$ (fails 2). B) $f(x,y)=y/x$. $f(x,0)=0/x=0$ (satisfies 1). $f(0,y)=y/0$ is undefined (satisfies 2). On a line $y=mx$, the slope is $f(x,mx) = (mx)/x = m$, which is constant (satisfies 3). This is the correct choice. C) $f(x,y)=xy$. $f(x,0)=0$ (satisfies 1). $f(0,y)=0$ (fails 2). D) $f(x,y)=x^2-y^2$. $f(x,0)=x^2$ (fails 1).

Question 2

1. The region where $y > x$. (correct answer)
2. The region where $y < x-1$.
3. The region between the lines $y=x$ and $y=x-1$.
4. The region where $y > x$ and $y < x-1$, which is an empty region.

Explanation: This problem requires a two-step analysis. First, we find the condition for the solution curves to be decreasing. A function is decreasing when its derivative is negative. So we need $dy/dx < 0$. For the given equation, this means $x-y < 0$, which simplifies to $y > x$. Second, we find the condition for the solution curves to be concave up. A function is concave up when its second derivative is positive. We find the second derivative using implicit differentiation: $d^2y/dx^2 = d/dx(x-y) = 1 - dy/dx$. Substituting the original equation, we get $d^2y/dx^2 = 1 - (x-y) = 1-x+y$. For the curve to be concave up, we need $d^2y/dx^2 > 0$, which means $1-x+y > 0$, or $y > x-1$. We need both conditions to be satisfied simultaneously: (1) $y > x$ and (2) $y > x-1$. Since the line $y=x$ is always above the line $y=x-1$, any point satisfying $y>x$ automatically satisfies $y>x-1$. Therefore, the region where both conditions hold is simply the region defined by $y > x$.

Question 3

1. $dy/dx = x^2y$
2. $dy/dx = xy^2$ (correct answer)
3. $dy/dx = x+y$
4. $dy/dx = x^2+y^2$

Explanation: Symmetry with respect to the y-axis means that if a solution curve has a slope $m$ at a point $(x,y)$, it must have a slope of $-m$ at the point $(-x,y)$. This is because reflecting a line segment with slope $m$ across the y-axis results in a line segment with slope $-m$. Therefore, the function $f(x,y)$ must satisfy the condition $f(-x, y) = -f(x, y)$ for all $(x,y)$ in the domain. This means that $f(x,y)$ must be an odd function with respect to $x$. Let's test the given options: A) $f(x,y) = x^2y$. $f(-x,y) = (-x)^2y = x^2y = f(x,y)$. This function is even in $x$. B) $f(x,y) = xy^2$. $f(-x,y) = (-x)y^2 = -xy^2 = -f(x,y)$. This function is odd in $x$. This is the correct choice. C) $f(x,y) = x+y$. $f(-x,y) = -x+y$. This is neither even nor odd in $x$. D) $f(x,y) = x^2+y^2$. $f(-x,y) = (-x)^2+y^2 = x^2+y^2 = f(x,y)$. This function is even in $x$.

Question 4

1. $c < 0$ (correct answer)
2. $c = 0$
3. $c > 0$
4. For all values of $c$

Explanation: The correct answer is A. Equilibrium solutions occur where $dy/dt = 0$. For the equation $dy/dt = y^4 - c$, we set $y^4 - c = 0$, which gives $y^4 = c$. We are looking for real-valued equilibrium solutions for $y$.

• If $c > 0$, then $y = \pm \sqrt[4]{c}$, which gives two distinct real equilibrium solutions.
• If $c = 0$, then $y^4 = 0$, which gives one equilibrium solution, $y = 0$.
• If $c < 0$, then $y^4 = c$ has no real solutions for $y$, because the fourth power of any real number is non-negative. Therefore, for $c < 0$, there are no equilibrium solutions, meaning the slope field has no points with a slope of zero.

Question 5

1. $f(-x, -y) = f(x, y)$ (correct answer)
2. $f(-x, -y) = -f(x, y)$
3. $f(-x, y) = -f(x, y)$
4. $f(x, -y) = f(x, y)$

Explanation: The correct answer is A. The slope of the solution curve at any point $(x, y)$ is given by $f(x, y)$. Symmetry with respect to the origin for a slope field means that the slope at $(-x, -y)$ is identical to the slope at $(x, y)$. Translating this into a condition on the function $f$, we get $f(-x, -y) = f(x, y)$. This is the definition of an even function with respect to the transformation $(x, y) \to (-x, -y)$. (B) implies that the slope at $(-x, -y)$ is the negative of the slope at $(x, y)$. (C) describes a condition for symmetry of the solution curves about the y-axis. (D) is a condition related to symmetry about the x-axis, where if $y(x)$ is a solution, then $-y(x)$ might also be a solution to a related equation.

Question 6

1. The line $y = 2x - 1$ (correct answer)
2. The line $y = 2x + 1$
3. The line $y = x$
4. The parabola $y = x^2 - 1$

Explanation: The correct answer is A. An isocline is a curve along which the slope, $dy/dx$, is constant. In this problem, we are looking for the isocline where the slope is $C=1$. We set the differential equation equal to this value: $dy/dx = 1$ $2x - y = 1$ Solving this equation for $y$ gives the equation of the isocline: $y = 2x - 1$. This is the equation of a line. Along this line, every segment of the slope field will have a slope of 1. Distractors represent common algebraic errors or misunderstandings of the definition of an isocline.

Question 7

1. $f(x, y)$ is a function of only $x$.
2. $f(x, y)$ is a function of only $y$. (correct answer)
3. The differential equation is linear in $y$.
4. The differential equation is separable.

Explanation: The correct answer is B. The statement 'all slope segments along any given horizontal line are parallel' means that for a fixed value of $y$, say $y=c$, the slope $dy/dx$ is constant for all values of $x$. This implies that the function $f(x, y)$ does not depend on $x$. Therefore, $f(x, y)$ must be a function of $y$ only, which is an autonomous differential equation. (A) is incorrect; if $f$ were a function of $x$ only, slopes would be constant along vertical lines. (C) is incorrect; for example, $dy/dx = y^2$ has this property but is not linear. (D) is incorrect; while an equation of the form $dy/dx = g(y)$ is separable, not all separable equations $dy/dx = g(y)h(x)$ have this property unless $h(x)$ is a constant.

Question 8

1. The first equation suggests solutions approach the line $y = -x - 1$ asymptotically, while the second suggests solutions approach $y = x - 1$, reflecting the different equilibrium line orientations
2. Both equations suggest similar long-term behavior since they both have zero-slope lines and differ only in the sign of one term, which creates minor variations in the slope field orientation
3. The first equation suggests solutions spiral outward from equilibrium points, while the second suggests solutions spiral inward, because the different signs create opposite rotational effects in the slope field
4. The first equation suggests exponential growth behavior for most solution curves, while the second suggests exponential decay, because the sign of the $y$ term determines whether solutions grow or decay exponentially (correct answer)

Explanation: When analyzing slope fields for linear differential equations, focus on how the signs of coefficients affect solution behavior over time. The key insight is recognizing how the $y$ term's sign determines whether solutions grow or decay exponentially. For $\frac{dy}{dx} = x + y$, the positive $y$ term creates a feedback effect where larger $y$ values produce steeper positive slopes, driving solutions to grow exponentially. Even when $x$ is negative, sufficiently large $y$ values dominate, causing runaway growth. For $\frac{dy}{dx} = x - y$, the negative $y$ term creates the opposite effect: as $y$ increases, the slope becomes less positive (or more negative), naturally restraining growth and leading to exponential decay toward an equilibrium. Option A incorrectly identifies the equilibrium lines. While both equations do have equilibrium solutions (where $\frac{dy}{dx} = 0$), the long-term behavior isn't about approaching these lines asymptotically in the way described. Option B completely misses the fundamental difference. The sign change in the $y$ term creates dramatically different behaviors, not "minor variations." Option C introduces spiraling behavior that doesn't exist in these linear systems. These equations produce exponential growth or decay along specific trajectories, not spiral patterns. Option D correctly identifies that the sign of the $y$ coefficient determines exponential behavior: positive coefficients lead to exponential growth, negative coefficients to exponential decay. Study tip: In linear differential equations of the form $\frac{dy}{dx} = ax + by$, the sign of the $y$ coefficient ($b$) is the primary indicator of long-term solution behavior—positive means growth, negative means decay.

Question 9

1. Regions far from the origin where both $|x|$ and $|y|$ are large, because the quadratic terms in the numerator dominate and create increasingly steep slope variations
2. Regions near the lines $y = \pm x$ where the numerator $x^2 - y^2$ approaches zero, because slopes transition from large positive to large negative values across these lines
3. Regions in the first and third quadrants where $xy > 0$, because the positive denominator amplifies the variability in the numerator, creating more sensitive slope changes
4. Regions near the coordinate axes where either $x$ or $y$ approaches zero, because the denominator $xy$ makes slopes approach infinity, creating rapid directional changes (correct answer)

Explanation: When analyzing slope fields, you need to identify where the differential equation's behavior becomes most extreme or unstable. For $\frac{dy}{dx} = \frac{x^2 - y^2}{xy}$, the most dramatic changes occur where the expression approaches infinity or undergoes rapid sign changes. The key insight is examining what happens as the denominator $xy$ approaches zero. When either $x$ or $y$ gets close to zero, the denominator becomes very small, causing the slope $\frac{dy}{dx}$ to become extremely large in magnitude. This creates regions where solution curves must make sharp turns or become nearly vertical, resulting in dramatic directional changes over small distances. Option A is incorrect because large values of $|x|$ and $|y|$ actually stabilize the slope field - both numerator and denominator grow, so the ratio doesn't become extreme. Option B misunderstands the behavior near $y = \pm x$. While the numerator does approach zero there, creating slopes near zero, this represents smooth transitions, not dramatic changes. The slopes change gradually from positive to negative. Option C incorrectly suggests that positive denominators amplify variability - the sign of $xy$ doesn't determine sensitivity to changes. Option D correctly identifies that near the coordinate axes, where $xy \to 0$, slopes approach $\pm\infty$, forcing solution curves to change direction rapidly over tiny distances. Strategy tip: For slope field analysis, always check where denominators approach zero - these create the most extreme behavior and dramatic directional changes in the field.

Question 10

1. The slope field exhibits periodic behavior with period $2\pi$ in both the $x$ and $y$ directions, creating a regular grid-like pattern of repeated slope orientations
2. The slope field has horizontal line segments (slope = 0) along the parallel lines $x + y = n\pi$ where $n$ is any integer, with slopes alternating between positive and negative in adjacent strips (correct answer)
3. The slope field shows radial symmetry around points where $x + y = \frac{\pi}{2} + n\pi$, with slopes pointing outward from these centers in all directions
4. The slope field has vertical asymptotes along lines $x + y = \frac{\pi}{2} + n\pi$ where the slope approaches infinity, creating discontinuous behavior in the field

Explanation: Since $\frac{dy}{dx} = \sin(x + y)$, the slope equals zero when $\sin(x + y) = 0$, which occurs when $x + y = n\pi$ for integer $n$. The slope is positive when $0 < x + y - n\pi < \pi$ and negative when $\pi < x + y - n\pi < 2\pi$, creating alternating strips. Choice A is wrong because the period in the slope field direction isn't simply $2\pi$ in both coordinate directions. Choice C incorrectly identifies maximum slope points as centers of radial symmetry. Choice D is wrong because $\sin(x + y)$ is bounded and never infinite.

Question 11

1. The claim is correct because $y = 2$ and $y = -2$ are equilibrium solutions where $\frac{dy}{dx} = 0$, and the uniqueness theorem prevents solution curves from intersecting (correct answer)
2. The claim is incorrect because while $\frac{dy}{dx} = 0$ at these lines, solution curves can still cross them since the derivative exists and is continuous everywhere
3. The claim is partially correct: curves cannot cross $y = 2$ but can cross $y = -2$ because the stability analysis shows different behavior at these equilibrium points
4. The claim is incorrect because these are not equilibrium solutions; when $y = \pm 2$, we have $\frac{dy}{dx} = \pm 4$, allowing curves to cross with nonzero slopes

Explanation: When $y = 2$, $\frac{dy}{dx} = 4 - 4 = 0$, and when $y = -2$, $\frac{dy}{dx} = 4 - 4 = 0$. These are equilibrium solutions (constant solutions) to the differential equation. By the uniqueness theorem for differential equations, since the function $f(x,y) = y^2 - 4$ and its partial derivative $\frac{\partial f}{\partial y} = 2y$ are continuous everywhere, solution curves cannot intersect. Choice B misunderstands that continuity doesn't allow crossing of distinct solutions. Choice C is wrong because both lines have the same equilibrium property. Choice D incorrectly calculates the derivative values.

Question 12

1. Solution curves are indeed strictly increasing and approach horizontal asymptotes as $x \to \infty$ because the large slopes eventually level off due to the $y^2$ term
2. Solution curves are strictly increasing but exhibit vertical asymptotes at finite $x$-values because the slopes become infinite in finite time, preventing curves from being defined for all $x$ (correct answer)
3. Solution curves oscillate with increasing amplitude as $x$ increases because the large slopes cause the curves to overshoot and create wave-like patterns
4. Solution curves are strictly increasing and have inflection points where $y^2 = 1$, causing changes in concavity that create S-shaped curves extending to infinity

Explanation: Since $\frac{dy}{dx} = 1 + y^2 \geq 1 > 0$, curves are indeed strictly increasing. However, as $y$ increases, $\frac{dy}{dx}$ grows without bound. This rapid growth means $y$ reaches infinity in finite time, creating vertical asymptotes. The solution is $y = \tan(x + C)$, which has vertical asymptotes. Choice A is wrong because curves don't approach horizontal asymptotes. Choice C is incorrect because there's no oscillation. Choice D misunderstands the concavity analysis - inflection points occur where the second derivative is zero, not where $y^2 = 1$.

Question 13

1. Along any ray $y = mx$ where $m$ is constant, all slope field segments have the same slope $m$, indicating that these rays are themselves solution curves to the differential equation (correct answer)
2. Along any ray $y = mx$ where $m$ is constant, the slope field segments have varying slopes that increase linearly with distance from the origin, creating a spiral pattern
3. Along any ray $y = mx$ where $m$ is constant, the slope field segments have slopes that approach $m$ asymptotically as the distance from origin increases, but never exactly equal $m$
4. Along rays from the origin, the slope field is undefined because the differential equation has a singularity at $(0,0)$, making analysis along these rays impossible

Explanation: For the differential equation $\frac{dy}{dx} = \frac{y}{x}$, along any ray $y = mx$ (where $x \neq 0$), we have $\frac{dy}{dx} = \frac{mx}{x} = m$. Since the slope of the ray $y = mx$ is also $m$, the slope field direction matches the ray direction exactly, confirming these rays are solution curves. Choice B is incorrect because the slopes are constant, not increasing. Choice C is wrong because the slopes exactly equal $m$, not approach it. Choice D misunderstands that while there's a singularity at the origin, the rays themselves (excluding the origin) are well-defined.

Question 14

1. Solution curves cross the line $y = x$ with slopes that increase as $x$ increases, since the slope field has value $2x$ along this line
2. Solution curves are tangent to the line $y = x$ at isolated points, creating a pattern of curves that approach but never cross this line
3. Solution curves cross the line $y = x$ with zero slope, then immediately curve away from the line in predictable directions based on the sign of $x$ (correct answer)
4. Solution curves follow the line $y = x$ exactly, since substituting $y = x$ into the differential equation yields $\frac{dy}{dx} = 0$

Explanation: When $y = x$, we have $\frac{dy}{dx} = x^2 - x^2 = 0$, so solution curves cross the line with zero slope. For points slightly above the line ($y = x + \epsilon$), $\frac{dy}{dx} = x^2 - (x + \epsilon)^2 \approx -2x\epsilon$, which is negative when $x > 0$ and positive when $x < 0$. This means curves curve away from the line after crossing it. Choice A incorrectly states the slope is $2x$ along $y = x$. Choice B is wrong because curves do cross the line. Choice D is incorrect because $y = x$ is not a solution to the differential equation.

Question 15

1. The slopes approach infinity, causing solution curves to become vertical as they near the line $x + y = 0$, which acts as a vertical asymptote for all solution curves
2. The slopes approach $-1$ uniformly along the line $x + y = 0$, and solution curves cross this line with slope $-1$ before continuing into the adjacent region
3. The differential equation is undefined along $x + y = 0$, creating a discontinuity in the slope field where no slope segments can be drawn (correct answer)
4. The slopes approach different finite limits depending on the direction of approach, with curves from different regions having distinct limiting behaviors as they near the line

Explanation: The differential equation $\frac{dy}{dx} = \frac{x - y}{x + y}$ has a denominator of zero when $x + y = 0$, making the equation undefined along this line. This creates a genuine discontinuity in the slope field. Choice A is incorrect because while slopes may become large near the line, the primary issue is that the equation is undefined, not that slopes approach infinity. Choice B is wrong because we cannot evaluate the limit of the slope as we approach the line. Choice D incorrectly suggests the slopes have well-defined directional limits when the expression is actually undefined.