Differential Equations: Problem Set 4

  1. Solve the nonlinear equation.

    \[x'=(t+x)^2\]

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

  2. Solve the second-order equation.

    \[x''+x'=3t\]

    Hint: Transform it into a first-order linear equation by making the substitution \(y=x'\).

  3. Consider the first-order linear ODE,

    \[x'+p(t)x=q(t)\]

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

    2. Now suppose that \(x_1(t)\) is a solution and \(x_2(t)\) is a solution to the homogeneous problem \(x'+p(t)x=0\). Show that \(x_1(t) + x_2(t)\) is a solution.

  4. 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.

  5. A differential equation of the form

    \[x'=a(t)x+g(t)x^n\]

    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: \(x'=\frac{2}{3t}x+\frac{2t}{x}\)

    3. Solve: \(t^2x'+2tx-x^3=0\)

  6. 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. \(N'=r(t)N\left(1-\frac{N}{K}\right)\)

    2. \(N'=rN\left(1-\frac{N}{K(t)}\right)\)

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

    \[M(t,x)+N(t,x)x'=0\]

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

    \[\frac{d}{dt}\psi (t,x)=M(t,x)+N(t,x)x'.\]

    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 \(M(t,x)+N(t,x)x'=0\) is exact if and only if \(M_x\) = \(N_t\).

    2. Solve: \(t^3+\frac{x}{t}+(x^2+\ln t)x'=0\)