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Differential Equations with Discontinuous Forcing Functions Calculator

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Example
Created on 2024-06-20Asked by Daniel Scott (Solvelet student)
Solve the differential equation dydx=1 \frac{dy}{dx} = 1 with y(0+)=2 y(0^+) = 2 and y(0−)=1 y(0^-) = 1 .

Solution

Step 1: Write the given differential equation: dydx=1 \frac{dy}{dx} = 1 . Step 2: Integrate both sides with respect to x x : ∫dydx dx=∫1 dx. \int \frac{dy}{dx} \, dx = \int 1 \, dx. Step 3: Perform the integration: y=x+C, y = x + C, where C C is the constant of integration. Step 4: Apply the initial conditions: y(0+)=2andy(0−)=1. y(0^+) = 2 \quad \text{and} \quad y(0^-) = 1. Step 5: The solution will have a jump discontinuity at x=0 x = 0 due to the different initial conditions. Step 6: Evaluate y(0+) y(0^+) and y(0−) y(0^-) to find C C : 2=0+C  ⟹  C=2. 2 = 0 + C \implies C = 2. 1=0+C  ⟹  C=1. 1 = 0 + C \implies C = 1. Step 7: Conclusion. Since the constant C C cannot take two different values, this problem does not have a solution. Solved on Solvelet with Basic AI Model
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1. Solve the initial value problem y′′+2y′+2y=5u(t−1) y'' + 2y' + 2y = 5u(t - 1) , where u(t) u(t) is the unit step function, subject to initial conditions y(0)=0 y(0) = 0 and y′(0)=0 y'(0) = 0 .2. Find the solution to the differential equation y′′+4y=δ(t−2) y'' + 4y = \delta(t - 2) , where δ(t) \delta(t) is the Dirac delta function, subject to initial conditions y(0)=0 y(0) = 0 and y′(0)=0 y'(0) = 0 .
DefinitionDifferential equations with piecewise constant forcing functions refer to differential equations in the form of a functional relationship between the unknown function and its derivatives. However, the system’s behaviour can be influenced by some random external inputs or stimuli known as forces, which are piecewise constant functions. These forces create discontinuities or jumps in the behaviour of the system; hence, it is challenging to analyze. Besides the generalized methods, laplace transforms, Heaviside functions or impulse responses is used to solve such differential equations. Formally, a differential equation involving a forcing function that is piecewise constant is denoted as; n\delta(t-0) Differential equations with discontinuous forcing functions play a vital role in transient response modeling or simulating sudden system changes. They can be applied in control theory, signal processing, or circuit analysis. For instance, a mass-spring-damper system response to an impulsive force can be represented using a differential equation. The forcing function used in the differential equation model is the Dirac delta function, representing the instantaneous application of the force.
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