Solving Initial Value Problems

Transforming derivatives turns a second-order initial value problem into algebra in s, which still works when the forcing term jumps or is piecewise.

The ODE Project, Solving Initial Value Problems

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13 exercises

Exercise 1

Solve the initial problems in Solving Initial Value Problems using the Laplace transform.

$y'' - 2y' - 3y = 3 e^{2t}$ , $y(0) = 1$ , $y'(0) = 0$

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Exercise 2

Solve the initial problems in Solving Initial Value Problems using the Laplace transform.

$y'' - y' - 2y = 4x^2$ , $y(0) = -1$ , $y'(0) = 1$

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Exercise 3

Solve the initial problems in Solving Initial Value Problems using the Laplace transform.

$\dfrac{d^2x}{dx^2} - 6 \dfrac{dx}{dt} + 25 x = 64e^{-t}$ , $x(0) = 1$ , $x'(0) = -2$

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Exercise 4

Solve the initial problems in Solving Initial Value Problems using the Laplace transform.

$y'' + 16y = 2 \sin 2t$ , $y(0) = 1$ , $y'(0) = 0$

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Exercise 5

Solve the initial problems in Solving Initial Value Problems using the Laplace transform.

$y'' + 16y = 2 \sin 4t$ , $y(0) = 1$ , $y'(0) = 0$

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Exercise 6

Solve the initial problems in Solving Initial Value Problems using the Laplace transform.

$y'' + 2y' + y = 2e^{-t}$ , $y(0) = -1$ , $y'(0) = 3$

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Exercise 7

Solve the initial problems in Solving Initial Value Problems using the Laplace transform.

$y'' + 6y' + 8y = \cos 3t$ , $y(0) = -2$ , $y'(0) = 1$

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Exercise 8

Solve the initial problems in Solving Initial Value Problems using the Laplace transform.

$u'' + \omega_0^2 y = \cos \omega t$ , $\omega^2 \neq \omega_0^2$ , $u(0) = 1$ , $u'(0) = -1$

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Exercise 9

Find the solution of the initial value problem where $g(t)$ is defined by

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Exercise 10

Let $y = y(t)$ and $y'(t)$ be piecewise continuous, exponentially bounded functions and assume that $y''$ is exponentially bounded.

Prove that for large values of $s$ , where $Y(s)$ is the Laplace transform of $y$ .
If $y = y(t)$ and all of its derivatives up to order $k - 1$ are piecewise continuous, exponentially bounded functions and $y^{(k)}$ is piecewise continuous, prove that

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Exercise 11

Prove this theorem.

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Exercise 12

Let $f(t)$ be a peicewise continuous function for $t \geq 0$ . In addition, suppose that $f(t)$ satisfies the condition $|f(t)| \leq M e^{ct}$ with $t \geq 0$ .

Prove that
Show that follows from (a).

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Exercise 13

Consider the linear system If we let ${\mathcal L}(u(s)) = U(s)$ and ${\mathcal L}(v(s)) = V(s)$ and take the Laplace transform of the two differential equations, we obtain

Verify that if we solve the system for $U(s)$ and $V(s)$ , we obtain
Find $u(t)$ and $v(t)$ .
Solve the linear system using Laplace transforms.
Solve the linear system using Laplace transforms.

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