Differential Equations: Problem Set 4

Solve the nonlinear equation.

$$x^{\prime }=(t+x{)}^2$$

Hint: Transform it into a separable equation by making the substitution $y=t+x$.

Solve the second-order equation.

$$x^{\prime \prime }+x^{\prime }=3t$$

Hint: Transform it into a first-order linear equation by making the substitution $y=x^{\prime }$.

Consider the first-order linear ODE,

$$x^{\prime }+p\left(t\right)x=q\left(t\right)$$

1. Let $x_1\left(t\right)$ and $x_2\left(t\right)$ be solutions. Show that $x_1\left(t\right)+x_2\left(t\right)$ is a solution if and only if $q\left(t\right)\equiv 0$.

2. Now suppose that $x_1\left(t\right)$ is a solution and $x_2\left(t\right)$ is a solution to the homogeneous problem $x^{\prime }+p\left(t\right)x=0$. Show that $x_1\left(t\right)+x_2\left(t\right)$ is a solution.

Initially a tank has 60 gallons of water. A brine solution with 1 pound of salt per gallon enters the (continuously stirred and well mixed) tank at a rate of 2 gallons per minute. Solution leaves the tank at a rate of 3 gallons per minute. Find an equation that gives the amount of salt in the tank at time t for all times beginning at the initial time up until the tank is empty.

A differential equation of the form

$$x^{\prime }=a\left(t\right)x+g\left(t\right)x^n$$

is called a Bernoulli equation.

1. Show that a Bernoulli equation can be transformed into a linear equation by making the substitution $y=x^{1-n}$ and then find a formula for the solution.

2. Solve: $x^{\prime }=\frac{2}{3t}x+\frac{2t}{x}$

3. Solve: $t^2{x}^{\prime }+2tx-x^3=0$

The following are two different generalizations of the logistic differential equation. Do you see why? In each case, solve the equations. Hint: they are Bernoulli equations.

1. $N^{\prime }=r\left(t\right)N\left(1-\frac{N}{K}\right)$

2. $N^{\prime }=rN\left(1-\frac{N}{K\left(t\right)}\right)$

(Bonus) The left hand side of the differential equation:

$$M(t,x)+N(t,x)x^{\prime }=0$$

looks like it might be a total derivative. It is a total derivative if there exists a function $\psi (t,x)$ such that,

$$\frac{d}{dt}\psi (t,x)=M(t,x)+N(t,x)x^{\prime }.$$

If we can find such a function, then the differential equation above is called an exact equation and we would have that:

$$\frac{d}{dt}\psi (t,x)=0,$$

which implies that the solution to the equation is given implicitly by $\psi (t,x)=C$.

1. Show that $M(t,x)+N(t,x)x^{\prime }=0$ is exact if and only if $M_x$ = $N_t$.

2. Solve: $t^3+\frac{x}{t}+(x^2+\ln t)x^{\prime }=0$