0.E: Introduction (Exercises)

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

Exercise \(\PageIndex{0.2.1}\)

Show that \(x = e^{4t}\) is a solution to \(x'''-12 x'' + 48 x' - 64 x = 0\).

Exercise \(\PageIndex{0.2.2}\)

Show that \(x = e^{t}\) is not a solution to \(x'''-12 x'' + 48 x' - 64 x = 0\).

Exercise \(\PageIndex{0.2.3}\)

Is \(y = \sin t\) a solution to \({\left( \frac{dy}{dt} \right)}^2 = 1 - y^2\)? Justify.

Exercise \(\PageIndex{0.2.4}\)

Let \(y'' + 2y' - 8y = 0\). Now try a solution of the form \(y = e^{rx}\) for some (unknown) constant \(r\). Is this a solution for some \(r\)? If so, find all such \(r\).

Exercise \(\PageIndex{0.2.5}\)

Verify that \(x = C e^{-2t}\) is a solution to \(x' = -2x\). Find \(C\) to solve for the initial condition \(x(0) = 100\).

Exercise \(\PageIndex{0.2.6}\)

Verify that \(x = C_1 e^{-t} + C_2 e^{2t}\) is a solution to \(x'' - x' -2 x = 0\). Find \(C_1\) and \(C_2\) to solve for the initial conditions \(x(0) = 10\) and \(x'(0) = 0\).

Exercise \(\PageIndex{0.2.7}\)

Find a solution to \({(x')}^2 + x^2 = 4\) using your knowledge of derivatives of functions that you know from basic calculus.

Exercise \(\PageIndex{0.2.8}\)

Solve:

1. \(\dfrac{dA}{dt} = -10 A, \quad A(0)=5\)
2. \(\dfrac{dH}{dx} = 3 H, \quad H(0)=1\)
3. \(\dfrac{d^2y}{dx^2} = 4 y, \quad y(0)=0, \quad y'(0)=1\)
4. \(\dfrac{d^2x}{dy^2} = -9 x, \quad x(0)=1, \quad x'(0)=0\)

Exercise \(\PageIndex{0.2.9}\)

Is there a solution to \(y' = y\), such that \(y(0) = y(1)\)?

Exercise \(\PageIndex{0.2.10}\)

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?

Exercise \(\PageIndex{0.2.11}\)

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.

1. What is \(s(0)\) and \(s(1)\).
2. Approximately how many years will it take for the salary to be \(10\) million.
3. Approximately how many years will it take for the salary to be \(20\) million.
4. Approximately how many years will it take for the salary to be \(30\) million.

Exercise \(\PageIndex{0.2.12}\)

Show that \(x = e^{-2t}\) is a solution to \(x'' + 4x' + 4x = 0\).

Answer

Compute \(x'=-2e^{-2t}\) and \(x''=4e^{-2t}\). Then \((4e^{-2t})+4(-2e^{-2t})+4(e^{-2t})=0\).

Exercise \(\PageIndex{0.2.13}\)

Is \(y = x^2\) a solution to \(x^2y'' - 2y = 0\)? Justify.

Answer

Yes.

Exercise \(\PageIndex{0.2.14}\)

Let \(xy'' - y' = 0\). Try a solution of the form \(y = x^r\). Is this a solution for some \(r\)? If so, find all such \(r\).

Answer

\(y=x^{r}\) is a solution for \(r=0\) and \(r=2\).

Exercise \(\PageIndex{0.2.15}\)

Verify that \(x=C_1e^t+C_2\) is a solution to \(x''-x' = 0\). Find \(C_1\) and \(C_2\) so that \(x\) satisfies \(x(0) = 10\) and \(x'(0) = 100\).

Answer

\(C_{1}=100\), \(C_{2}=-90\)

Exercise \(\PageIndex{0.2.16}\)

Solve \(\frac{d\varphi}{ds} = 8 \varphi\) and \(\varphi(0) = -9\).

Answer

\(\varphi =-9e^{8s}\)

Exercise \(\PageIndex{0.2.17}\)

Solve:

1. \(\dfrac{dx}{dt} = -4x, \quad x(0)=9\)
2. \(\dfrac{d^2x}{dt^2} = -4x, \quad x(0)=1, \quad x'(0)=2\)
3. \(\dfrac{dp}{dq} = 3 p, \quad p(0)=4\)
4. \(\dfrac{d^2T}{dx^2} = 4 T, \quad T(0)=0, \quad T'(0)=6\)

Answer

1. \(x=9e^{-4t}\)
2. \(x=\cos (2t)+\sin (2t)\)
3. \(p=4e^{3q}\)
4. \(T=3\sinh (2x)\)

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

Exercise \(\PageIndex{0.3.1}\)

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?

1. \(\displaystyle \sin(t) \frac{d^2 x}{dt^2} + \cos(t) x = t^2\)
2. \(\displaystyle \frac{\partial u}{\partial x} + 3 \frac{\partial u}{\partial y} = xy\)
3. \(\displaystyle y''+3y+5x=0, \quad x''+x-y=0\)
4. \(\displaystyle \frac{\partial^2 u}{\partial t^2} + u\frac{\partial^2 u}{\partial s^2} = 0\)
5. \(\displaystyle x''+tx^2=t\)
6. \(\displaystyle \frac{d^4 x}{dt^4} = 0\)

Exercise \(\PageIndex{0.3.2}\)

If \(\vec{u} = (u_1,u_2,u_3)\) is a vector, we have the divergence \(\nabla \cdot \vec{u} = \frac{\partial u_1}{\partial x} + \frac{\partial u_2}{\partial y} + \frac{\partial u_3}{\partial z}\) and curl \(\nabla \times \vec{u} = \Bigl( \frac{\partial u_3}{\partial y} - \frac{\partial u_2}{\partial z} , ~ \frac{\partial u_1}{\partial z} - \frac{\partial u_3}{\partial x} , ~ \frac{\partial u_2}{\partial x} - \frac{\partial u_1}{\partial y} \Bigr)\). Notice that curl of a vector is still a vector. Write out Maxwell’s equations in terms of partial derivatives and classify the system.

Exercise \(\PageIndex{0.3.3}\)

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 \(F(y',y) = 0\).

Exercise \(\PageIndex{0.3.4}\)

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.

Exercise \(\PageIndex{0.3.5}\)

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?

1. \(\displaystyle \frac{\partial^2 v}{\partial x^2} + 3 \frac{\partial^2 v}{\partial y^2} = \sin(x)\)
2. \(\displaystyle \frac{d x}{dt} + \cos(t) x = t^2+t+1\)
3. \(\displaystyle \frac{d^7 F}{dx^7} = 3F(x)\)
4. \(\displaystyle y''+8y'=1\)
5. \(\displaystyle x''+tyx'=0, \quad y''+txy = 0\)
6. \(\displaystyle \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial s^2} + u^2\)

Answer

1. PDE, equation, second order, linear, nonhomogeneous, constant coefficient.
2. ODE, equation, first order, linear, nonhomogeneous, not constant coefficient, not autonomous.
3. ODE, equation, seventh order, linear, homogeneous, constant coefficient, autonomous.
4. ODE, equation, second order, linear, nonhomogeneous, constant coefficient, autonomous.
5. ODE, system, second order, nonlinear.
6. PDE, equation, second order, nonlinear.

Exercise \(\PageIndex{0.3.6}\)

Write down the general zeroth order linear ordinary differential equation. Write down the general solution.

Answer

equation: \(a(x)y=b(x)\), solution: \(y=\frac{b(x)}{a(x)}\).

Exercise \(\PageIndex{0.3.7}\)

For which \(k\) is \(\frac{dx}{dt}+x^k = t^{k+2}\) linear. Hint: there are two answers.

Answer

\(k=0\) or \(k=1\)