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Differential Equation Solution via Laplace Transform
Published on May 29, 2025
Calculus
Find the solution to a differential equation given its Laplace transform. The solution involves inverse Laplace transforms and the time-shifting property.
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Published on May 29, 2025
Find the solution to a differential equation given its Laplace transform. The solution involves inverse Laplace transforms and the time-shifting property.
This calculus problem involves evaluating the integral of (e^x + 1) / (e^x - 1) dx. The solution uses substitution and partial fraction decomposition to find the antiderivative.
Solve the second-order linear differential equation y'' - y = 0 using the Method of Undetermined Power Series Coefficients, resulting in a solution involving hyperbolic functions.
The provided text contains a detailed solution to various integration problems, including trigonometric substitutions, partial fraction decomposition, and integration by parts. It also demonstrates the use of properties of definite integrals to simplify calculations.
The Laplace transform of the piecewise function f(t) is calculated using Heaviside step functions and the time-shifting property of Laplace transforms. The result is expressed in terms of the exponential function and the reciprocal of s squared.
The function f(x,y) = -2x^2 + 3xy^2 - 3y^2 + 4 has a local maximum value of 4 at (0,0).
The double integral, using polar coordinates and integration by parts, evaluates to π(e^2 + 1).
The solution attempts to derive a reduction formula for the integral \(\int_0^\infty \frac{x}{\cosh^n(x)} dx\) using integration by parts and Feynman's trick, but ultimately finds that expressing the integral in terms of hypergeometric functions is a more suitable approach.
The integral \int_{0}^{\infty} \frac{x}{(\cosh x)^2} dx evaluates to ln(2).
To evaluate the limit, direct substitution results in an indeterminate form (0/0). Factoring the numerator reveals a common factor of (x-3), which cancels with the denominator. Substituting x=3 into the simplified expression yields a final answer of 58.
The integrals were evaluated using trigonometric substitution, partial fraction decomposition, and substitution. Integral c) was solved using trigonometric substitution, resulting in x/(√(1-x^2)) + C. Integral d) diverged to infinity. Integral e) was solved using partial fraction decomposition, leading to a solution involving natural logarithms and a constant term.