Chapter D1, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a.    The differential equation y′ + 2y = t is first-order, linear, and separable.

b.    The differential equation y′y = 2t² is first-order, linear, and separable.

c.    The function y = t + 1/t satisfies the initial value problem ty′ + y = 2t, y(1) = 2.

d.    The direction field for the differential equation y′(t) = t + y(t) is plotted in the ty-plane.

e.    Euler’s method gives the exact solution to the initial value problem y′ = ty², y(0) = 3 on the interval [0, a] provided a is not too large.

To determine

Whether the given statement is true or false.

Answer to Problem 1RE

The given statement is False.

Explanation of Solution

The given statement is “The differential equation $y^{\prime }\left(t\right)+2y=t$ is first order, linear, and separable.”

It is known that order of a differential equation is the highest derivative present in the given differential equation.

Therefore, from the given differential equation observe that the highest order of derivative is 1.

Thus, the given differential equation is a first order differential equation.

Also, note that for a differential equation to be linear, the equation must not have products or quotients of y and its derivatives.

Thus, it can be concluded that the given differential equation is linear.

Now, check whether the given differential equation is separable or not.

$\begin{array}{l}{y}^{\prime }\left(t\right)+2y=t\\ \frac{dy}{dt}+2y=t\\ \frac{dy}{dt}=t-2y\\ dy=\left(t-2y\right)dt\end{array}$

From the above equation, note that the given differential equation is not separable as the variables cannot be separated further.

Therefore, the equation is in first order, linear but not separable.

Thus, the statement is false.

To determine

Whether the given statement is true or false.

Answer to Problem 1RE

The statement is False.

Explanation of Solution

The given statement is “The differential equation $y^{\prime }y=2t^2$ is first order, linear and separable.”

It is known that order of a differential equation is the highest derivative present in the given differential equation.

Therefore, from the given differential equation observe that the highest order of derivative is 1.

Thus, the given differential equation is a first order differential equation.

Also, note that for a differential equation to be linear, the equation must not have products or quotients of y and its derivatives.

Thus, from the given differential equation note that the equation consists of the product of the variable y and its derivatives.

Thus, the equation is not linear.

Now, check whether the given differential equation is separable or not.

$\begin{array}{l}{y}^{\prime }y=2t^2\\ \frac{dy}{dt}y=2t^2\\ ydy=2t^{2}dt\end{array}$

From the above equation, observe that the variables can be separated.

Therefore, the equation is in first order, separable but not linear.

Thus, the statement is False.

To determine

Whether the given statement is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The given statement is “The function $y=t+\frac{1}{t}$ satisfies the initial value problem $t{y}^{\prime }+y=2t,y\left(1\right)=2$.”

Take derivative on both sides of the equation $y=t+\frac{1}{t}$ as shown below.

$\begin{array}{l}y=t+\frac{1}{t}\\ y^{\prime }=1-\frac{1}{t^2}\end{array}$

Now, substitute the value of $y^{\prime }$ in the given initial value problem.

$\begin{array}{l}t{y}^{\prime }+y=2t\\ t\left(1-\frac{1}{t^2}\right)+y=2t\\ t-t^{-1}+\left(t+t^{-1}\right)=2t\\ 2t=2t\end{array}$

Therefore, the function $y=t+\frac{1}{t}$ satisfies the initial value problem $t{y}^{\prime }+y=2t$.

Thus, the statement is true.

To determine

Whether the direction field for the differential equation $y^{\prime }\left(t\right)=t+y\left(t\right)$ is plotted in the $ty$-plane.

Answer to Problem 1RE

The statement “The direction field for the differential equation $y^{\prime }\left(t\right)=t+y\left(t\right)$ is plotted in the ty-plane” is $\underset{\_}{\text{True}}$.

Explanation of Solution

The given differential equation is $y^{\prime }\left(t\right)=t+y\left(t\right)$.

Note that the notation $f\left(t,y\right)$ is an expression involving the independent variable t and the unknown solution y.

Also, for the differential equation at each point $\left(t,y\right)$ of the solution curve, the slope of the curve is $y^{\prime }\left(t\right)=t+y\left(t\right)$.

It is known that a direction field is a picture that shows the slope of the solution at $ty$-plane.

Therefore the direction field for the differential equation $y^{\prime }\left(t\right)=t+y\left(t\right)$ is plotted in the $ty$-plane.

To determine

Whether the given statement is true or false.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The given statement is “The Euler’s method gives the exact solution to the initial value problem $y^{\prime }=t{y}^2,y\left(0\right)=3$ on the interval $\left[0,a\right]$, where a is not too large.”

The given initial value problem is $y^{\prime }=t{y}^2,y\left(0\right)=3$.

It is known that the direction fields are the basis for many Computer based methods for approximating solutions of a differential equation.

Also, the exact solution of the initial value problem at grid points is $y\left(t_k\right)$, for $k=0,1,\mathrm{2....}N$ which is generally unknown.

Therefore, the goal is to compute a set of approximations to the exact solution at the grid points.

Therefore, the given assumption is false.

Thus, the statement is false.