MATH 4038

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Last updated 6:25 PM on 9/17/26
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24 Terms

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What is a solution curve?

After solving a differential equation and you get a function, it is the graph of that function

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What if a derivative?

How fast something changes over time

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What is a function?

a rule with an input that has one output

<p>a rule with an input that has one output </p>
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What is the Independent variable?

The input.

Usually x or t

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Ordinary Differential Equation (ODE)?

A differential equation with one independent variable.

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What is a Partial Differential Equation (PDE)?

A differential equation with more than one independent variable

<p>A differential equation with more than one independent variable </p>
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What is a Linear differential equation?

y and its derivatives appear only to the first power

<p><span>y</span> and its derivatives appear only to the first power </p>
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What is a trivial solution?

The zero solution

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Explicit solution?

y is by itself

<p><span>y</span> is by itself </p>
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Implicit solution?

x and y are mixed together

<p>x and y are mixed together </p>
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Family of solutions?

A group of solutions containing constants

<p>A group of solutions containing constants </p>
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Two-parameter family?

A solution with two arbitrary constants

<p>A solution with two arbitrary constants</p>
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What is an autonomous differential equation?

A differential equation where x does not appear.

<p>A differential equation where x does <strong>not</strong> appear. </p>
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What is a nonautonomous differential equation?

A differential equation where x does appear.

<p>A differential equation where x does appear. </p>
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What is a critical point (equilibrium point)?

A value of y where


Its where the solution stays constant at that point

<p>A value of y where </p><p></p><p>Its where the solution stays constant at that point</p>
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When do solution arrows point up?

where the solution increases

<p>where the solution increases</p>
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Can a solution cross a critical point?

No. Solutions stay in their region.

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What is a relative maximum or minimum?

A turning point where a solution changes direction.

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What is a separable differential equation?

A differential equation where the x's and y's can be separated onto opposite sides, and then you can take the derivative

<p>A differential equation where the x's and <span>y</span>'s can be separated onto opposite sides, and then you can take the derivative </p>
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What is a homogeneous differential equation?

A linear DE w/ f(x) = 0

<p>A linear DE w/ f(x) = 0</p>
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What is reduction of order?

A method used to find a second solution y2y_2y2​ when one solution y1y_1y1​ is already known.

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What is a Cauchy-Euler equation?

A differential equation where the power of xxx matches the derivative order x2 goes with y′′, x goes with y′


<p>A differential equation where the power of xxx matches the derivative order x<span>2</span> goes with y′′, x goes with y′</p><p></p>
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What is an auxiliary equation?

The equation involving only mmm that is obtained after substituting y=xm into a Cauchy-Euler equation.

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