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Ordinary Differential Equations 1 Flashcards | StudBud
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Ordinary Differential Equations 1
Ordinary Differential Equations 1
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1
y'+x^2y=x^3+1. Find the integrating factor.
e^{\left(\frac{x^2}{2}\right)}
e^{\left(\frac{x^3}{3}\right)}
e^{x^2}
e^{\left(\frac{x^4}{4}\right)}
2
For \left(x^2+y^2\right)dx=\left(xy-y^2\right)dy, find the appropriate substitution to use.
u=y^{2-2}=y^0
u=\frac{x}{y}
u=\frac{y}{x}
u=y^{1-2}=\frac{1}{y}
3
Find the appropriate partial derivatives to use for 2xydx=\left(x^2-1\right)dy
\frac{df}{dx}=x^2-1;\frac{df}{dy}=2xy
\frac{df}{dx}=2y;\frac{df}{dy}=2x
\frac{df}{dx}=2y-2x;\frac{df}{dy}=2x
\frac{df}{dx}=2xy; \frac{df}{dy}=x^2-1
4
What do differential equations do?
Give derivatives new purposes and applications
Relate a function to its derivatives
Make integration easier
Find the derivatives of functions
5
Identify which is
not
a trait of this differential equation. 5y''-2y'+5y=0
Homogeneous
Linear
Second-order
Bernoulli
6
Verify whether y=3t+t^2 is a family of solutions for ty'-y=t^2. Give a reason.
No; t\left(3t+t^2\right)-3t-t^2\ne t^2
Yes; t\left(3+2t\right)-3t-t^2=t^2
No; t\left(\frac{3t^2}{2}+\frac{t^3}{3}\right)-\left(3t+t^2\right)\ne t^2
No; t\left(3+2t\right)-3t+t^2\ne t^2
7
For \left(x^2+y^2\right)dx=\left(xy-x^2\right)dy, find the appropriate substitution to use.
u=\frac{x}{y}
u=y^{2-2}=y^0
u=y^{1-2}=\frac{1}{y}
u=\frac{y}{x}
8
y'+x^2y=x^3+1. Solve for y.
y=e^{\left(\frac{x^3}{3}\right)}\left(x^3+1\right)
y=x+e^{\left(-\frac{x^3}{3}\right)}+c
y=x+ce^{-\left(\frac{x^3}{3}\right)}
y=ce^{\left(\frac{x^3}{3}\right)}
9
Finish solving 2xydx=\left(x^2-1\right)dy
y=\frac{1}{x^2-1}+c
y=\frac{1}{x^2+1}+c
y=\frac{c}{x^2+1}
y=\frac{c}{x^2-1}
10
Find the appropriate substitution for x\frac{dy}{dx}+y=x^2y^2.
u=y^{1-2}=\frac{1}{y}
u=\frac{y}{x}
u=y^{0-2}=\frac{1}{y^2}
IF:e^{-x}
11
For x^2y''-7xy'+15xy=0, find all values of
m
such that y=x^m is a valid solution.
0
\frac{\left(49\pm\sqrt{60}\right)}{2}
3 and 5
-3 and -5
12
Identify which of these linear differential equations is non-homogeneous.
xy''+xy'+x^2y=0
y''+5y'=xy
y'''+xy''+x^2y'+x^3y=0
y''-4y=\sin\left(x\right)
13
\frac{dy}{dx}=\frac{e^x}{\sin\left(y\right)}
y=\sin^{-1}\left(e^x\right)
y=\frac{1}{\cos\left(-e^x+c\right)}
y=\cos^{-1}\left(e^{-x}+c\right)
y=\cos^{-1}\left(-e^x+c\right)
14
Represent Newton's law of cooling as a DE. Use M as the temperature of surroundings and T as the object's temperature.
\frac{dT}{dt}=-k\left(T-M\right)
\frac{dT}{dt}=k\left(T-M\right)
\frac{dT}{dt}=k\left(M-T\right)
\frac{dT}{dt}=-k\left(M-T\right)
15
Which one is a Bernoulli DE?
x\frac{dy}{dx}+y=x^2y^2
y'+x^2y=x^3+1
\left(x^2+y^2\right)dx+\left(x^2-xy\right)dy=0
2xydx=\left(x^2-1\right)dy
16
Which model is appropriate for "the population of a town increases at a rate proportional to its population"?
\frac{dP}{dt}=\frac{k}{P}
\frac{dP}{dt}=kP\left(1-P\right)
\frac{dP}{dt}=\frac{kP^2}{2}
\frac{dP}{dt}=kP
17
Which calculus rule was applied in reverse during the previous slide's proceedings?
Power rule
Chain rule
Quotient rule
Product rule
18
Given linear second-order DE y''-4y'+4y=0 and solution y=e^{2x}, find a second solution.
xe^{2x}
xe^{2x}+e^{2x}
e^{-2x}
ce^{2x}
19
Solve x\frac{dy}{dx}+y=x^2y^2.
y=\frac{1}{c\left(x-x^2\right)}
y=\frac{c}{x-x^2}
y=\frac{1}{x-cx^2}
y=\frac{1}{cx-x^2}
