0.E: Introduction (Exercises)
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title="0.2: Introduction to Differential Equations" href="/Bookshelves/Differential_Equations/Book:_Differential_Equations_for_Engineers_(Lebl)/0:_Introduction/0.2:_Introduction_to_Differential_Equations">Introduction to Differential Equations
Show that x=e4t is a solution to x‴−12x″+48x′−64x=0.
Show that x=et is not a solution to x‴−12x″+48x′−64x=0.
Is y=sint a solution to (dydt)2=1−y2? Justify.
Let y″+2y′−8y=0. Now try a solution of the form y=erx for some (unknown) constant r. Is this a solution for some r? If so, find all such r.
Verify that x=Ce−2t is a solution to x′=−2x. Find C to solve for the initial condition x(0)=100.
Verify that x=C1e−t+C2e2t is a solution to x″−x′−2x=0. Find C1 and C2 to solve for the initial conditions x(0)=10 and x′(0)=0.
Find a solution to (x′)2+x2=4 using your knowledge of derivatives of functions that you know from basic calculus.
Solve:
- dAdt=−10A,A(0)=5
- dHdx=3H,H(0)=1
- d2ydx2=4y,y(0)=0,y′(0)=1
- d2xdy2=−9x,x(0)=1,x′(0)=0
Is there a solution to y′=y, such that y(0)=y(1)?
The population of city X was 100 thousand 20 years ago, and the population of city X was 120 thousand 10 years ago. Assuming constant growth, you can use the exponential population model (like for the bacteria). What do you estimate the population is now?
Suppose that a football coach gets a salary of one million dollars now, and a raise of 10% every year (so exponential model, like population of bacteria). Let s be the salary in millions of dollars, and t is time in years.
- What is s(0) and s(1).
- Approximately how many years will it take for the salary to be 10 million.
- Approximately how many years will it take for the salary to be 20 million.
- Approximately how many years will it take for the salary to be 30 million.
Show that x=e−2t is a solution to x″+4x′+4x=0.
- Answer
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Compute x′=−2e−2t and x″=4e−2t. Then (4e−2t)+4(−2e−2t)+4(e−2t)=0.
Is y=x2 a solution to x2y″−2y=0? Justify.
- Answer
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Yes.
Let xy″−y′=0. Try a solution of the form y=xr. Is this a solution for some r? If so, find all such r.
- Answer
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y=xr is a solution for r=0 and r=2.
Verify that x=C1et+C2 is a solution to x″−x′=0. Find C1 and C2 so that x satisfies x(0)=10 and x′(0)=100.
- Answer
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C1=100, C2=−90
Solve dφds=8φ and φ(0)=−9.
- Answer
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φ=−9e8s
Solve:
- dxdt=−4x,x(0)=9
- d2xdt2=−4x,x(0)=1,x′(0)=2
- dpdq=3p,p(0)=4
- d2Tdx2=4T,T(0)=0,T′(0)=6
- Answer
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- x=9e−4t
- x=cos(2t)+sin(2t)
- p=4e3q
- T=3sinh(2x)
title="0.3: Classification of Differential Equations" href="/Bookshelves/Differential_Equations/Book:_Differential_Equations_for_Engineers_(Lebl)/0:_Introduction/0.3:_Classification_of_Differential_Equations">Classification of Differential Equations
Classify the following equations. Are they ODE or PDE? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant coefficient? If it is an ODE, is it autonomous?
- sin(t)d2xdt2+cos(t)x=t2
- ∂u∂x+3∂u∂y=xy
- y″+3y+5x=0,x″+x−y=0
- ∂2u∂t2+u∂2u∂s2=0
- x″+tx2=t
- d4xdt4=0
If →u=(u1,u2,u3) is a vector, we have the divergence ∇⋅→u=∂u1∂x+∂u2∂y+∂u3∂z and curl ∇×→u=(∂u3∂y−∂u2∂z, ∂u1∂z−∂u3∂x, ∂u2∂x−∂u1∂y). Notice that curl of a vector is still a vector. Write out Maxwell’s equations in terms of partial derivatives and classify the system.
Suppose F is a linear function, that is, F(x,y)=ax+by for constants a and b. What is the classification of equations of the form F(y′,y)=0.
Write down an explicit example of a third order, linear, nonconstant coefficient, nonautonomous, nonhomogeneous system of two ODE such that every derivative that could appear, does appear.
Classify the following equations. Are they ODE or PDE? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant coefficient? If it is an ODE, is it autonomous?
- ∂2v∂x2+3∂2v∂y2=sin(x)
- dxdt+cos(t)x=t2+t+1
- d7Fdx7=3F(x)
- y″+8y′=1
- x″+tyx′=0,y″+txy=0
- ∂u∂t=∂2u∂s2+u2
- Answer
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- PDE, equation, second order, linear, nonhomogeneous, constant coefficient.
- ODE, equation, first order, linear, nonhomogeneous, not constant coefficient, not autonomous.
- ODE, equation, seventh order, linear, homogeneous, constant coefficient, autonomous.
- ODE, equation, second order, linear, nonhomogeneous, constant coefficient, autonomous.
- ODE, system, second order, nonlinear.
- PDE, equation, second order, nonlinear.
Write down the general zeroth order linear ordinary differential equation. Write down the general solution.
- Answer
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equation: a(x)y=b(x), solution: y=b(x)a(x).
For which k is dxdt+xk=tk+2 linear. Hint: there are two answers.
- Answer
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k=0 or k=1