1.2E: Exercises for Section 1.2

Exercise \(\PageIndex{1}\)

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. \(\displaystyle \frac{d^4 x}{dt^4} = 0\)

Note: For a nonlinear equation, homogeneous /nonhomogeneous, constant coefficient does not apply.

Answer

1. ODE, equation, second order, nonlinear, autonomous.
2. PDE, equation, first order, linear, nonhomogeneous, constant coefficient.
3. ODE, system, second order, linear, homogeneous, constant coefficient, autonomous.
4. PDE, equation, second order, nonlinear.
5. ODE, equation, second order, nonlinear, non-autonomous.
6. ODE, equation, fourth order, linear, homogeneous, constant coefficients, autonomous.

Exercise \(\PageIndex{2}\)

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 \(\PageIndex{3}\)

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 \(\PageIndex{4}\)

Write down an explicit example of a third order, linear, nonconstant coefficient, nonautonomous, nonhomogeneous system of two ODEs such that every derivative that could appear, does appear.

Answer

Equation1: \( \frac{d^3 y}{dx^3} + x\frac{d^2 y}{dx^2}+\frac{dy}{dx}+\sin(x)y = 4x \).

Equation2: \(\frac{d^3 y}{dx^3} + x\frac{d^2 y}{dx^2}+\frac{dy}{dx}+xy = e^x\). 

Exercise \(\PageIndex{5}\)

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 \(\PageIndex{6}\)

Write down the general \(0^{th}\) 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 \(\PageIndex{7}\)

For which \(k\) is \(\frac{dx}{dt}+x^k = t^{k+2}\) linear?

Hint: there are two answers.

Answer

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

Exercise \(\PageIndex{8}\)

Write the degree and the order of the equations

1. \( \left(\frac{d^2 y}{dx^2}\right)^4+\frac{dy}{dx} = 4 \)
2. \( \left(\frac{d^3 y}{dx^3}\right)^2+\left(\frac{d^2 y}{dx^2}\right)^3+\frac{dy}{dx} = 5x \)
3. \( \left(\frac{d^4 y}{dx^4}\right)^3+\left(\frac{d^2 y}{dx^2}\right)^3+\frac{dy}{dx}= 0 \)
4. \( \left(\frac{dy}{dx}\right)^3+ \left(\frac{d^4 y}{dx^4}\right)^2= 8y \)

Hint: For finding the degree, look for the exponent of the highest derivative in the equation.

Answer

1. \( Degree= 4, Order = 2 \)
2. \( Degree= 2, Order = 3 \)
3. \( Degree= 3, Order = 4 \)
4. \( Degree= 2, Order = 4 \)

Exercise \(\PageIndex{9}\)

Write a second-order, nonhomogeneous, nonlinear, non-autonomous ODE. 

Answer

\( \frac{d^2x}{d^2t} + \cos(t) x = t^2+t+1\).

Exercise \(\PageIndex{10}\)

For the given differential equation,

1. \( \left(\frac{d^2x}{d^2y} \right)^2+ xy^2 = 4\) 
2. \(\displaystyle y''+3y+5x=5 \) 

Answer the following questions and explain.

1. Order of the equation
2. Degree of the equation
3. Is it ODE
4. Is it homogeneous/nonhomogeneous valid?
5. Linear?
6. Constant coefficients
7. Autonomous

Answer

1.  
   1. Second order: highest derivative in the equation.
   2.  2: The degree of a differential equation is the power of the highest order derivative after removing radicals and fractions
   3. No, An equation is non-linear if the dependent variable or its derivatives are raised to a power other than 1.
   4. Non homogeneous
   5. No, The coefficient of the term \(x\) is \( y^2 \).
   6. Not valid, as it is not linear.
   7. Non-autonomous: An equation is autonomous if the independent variable \( y \) does not appear explicitly in the equation. Since \( y^2 \) is present, it is non-autonomous.
2.  
   1. Second order: highest derivative in the equation.
   2. 1, Power of the highest order derivative
   3. Yes, Powers of y or its derivatives are 1
   4. Non homogeneous
   5. Yes, It is of the form \( a_2(t) x'' + a_1(t) x' + a_0(t) x = g(x) \) where g(x) =5 but not 0.
   6. Yes, it is of the form \( a_2(x) y'' + a_1(x) y' + a_0(x) y =g(x)  \). Where \( a_2=1, a_1(t) = 0, a_0(t) = 3 \) and \(g(x)= -5x+5 \)
   7. Non-autonomous: The independent variable \( x \) appears explicitly in the term \( x \).