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Solving the Differential Equation (d^2+2d-3)y=0
In this article, we will explore the solution to the differential equation (d^2+2d-3)y=0. This is a second-order linear homogeneous differential equation, and we will use the characteristic equation to find the general solution.
The Characteristic Equation
To solve the differential equation, we need to find the characteristic equation. The characteristic equation is a quadratic equation in the form of:
r^2 + 2r - 3 = 0
This equation can be factorized as:
(r + 3)(r - 1) = 0
This gives us two possible values for r:
r = -3 and r = 1
The General Solution
Using the values of r, we can write the general solution to the differential equation as:
y = c1e^(-3x) + c2e^(x)
where c1 and c2 are arbitrary constants.
Interpretation of the Solution
The general solution consists of two parts: c1e^(-3x) and c2e^(x). The first part, c1e^(-3x), represents a decaying exponential function, while the second part, c2e^(x), represents a growing exponential function.
The arbitrary constants c1 and c2 can be determined by applying the initial conditions of the problem. For example, if the initial condition is y(0) = 1 and y'(0) = 2, we can substitute these values into the general solution to find the specific values of c1 and c2.
Conclusion
In conclusion, the solution to the differential equation (d^2+2d-3)y=0 is a combination of decaying and growing exponential functions. The general solution is y = c1e^(-3x) + c2e^(x), where c1 and c2 are arbitrary constants that can be determined by applying the initial conditions of the problem.
