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Determine the general solution of y+λ2y=m=1Namsinmπt  where λ>0 and λmπ for m=1,,N

Short Answer

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Answer: The general solution for the given inhomogeneous equation is y(t)=Acos(λt)+Bsin(λt)+m=1Namλ2m2π2sinmπt.

Step by step solution

01

Find the complementary solution for the Homogeneous equation

The given differential equation is inhomogeneous, so let's start by finding the complementary solution for the homogeneous equation: y+λ2y=0 This is a second order linear homogeneous differential equation with constant coefficients and has the auxiliary equation: r2+λ2=0 Solving this equation, we get: r=±λi So, the complementary solution (the general solution for the homogeneous) is given by: yc(t)=Acos(λt)+Bsin(λt) Where A and B are constants.
02

Finding a particular solution for the inhomogeneous equation

Now, we need to find a particular solution for the inhomogeneous equation. Since the given inhomogeneous equation is a sum of sines, we can find the solution for each independent term and then sum them up. Consider a single term in the inhomogeneous source: amsinmπt We can find a particular solution for this term by suggesting a solution of the form: yp(t)=Cmsinmπt Plug this solution into our inhomogeneous equation: yp+λ2yp=amsinmπt We get: (Cmm2π2+λ2Cm)sinmπt=amsinmπt Notice here that λ2m2π2, so we can solve for Cm without encountering division by zero: Cm=amλ2m2π2 Thus, yp(t)=amλ2m2π2sinmπt
03

Summing up the particular solutions

Now, sum up the particular solutions for all terms: yp(t)=m=1Namλ2m2π2sinmπt
04

Combine complementary and particular solutions

Finally, we'll combine the complementary and particular solutions to find the general solution of the inhomogeneous equation: y(t)=Acos(λt)+Bsin(λt)+m=1Namλ2m2π2sinmπt This is the general solution of the given differential equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complementary Solution
In the context of differential equations, the complementary solution represents the part of the solution that solves the associated homogeneous equation. This homogeneous equation is obtained by setting the function that accounts for any external forces (or inhomogeneous terms) to zero. For instance, given the equation y+λ2y=0, we only focus on the left-hand side, deliberately ignoring non-homogeneous or forced components.

To find the complementary solution for y+λ2y=0, we employ methods that solve linear homogeneous differential equations with constant coefficients. The auxiliary equation r2+λ2=0 guides us in figuring out the characteristic values, which are r=±λi. These complex roots imply an oscillatory form in the solution, resulting in a complementary solution of:
  • yc(t)=Acos(λt)+Bsin(λt)
Here, A and B are arbitrary constants, determined by initial conditions or boundary values if specified.
Homogeneous Equation
A homogeneous equation typically refers to a differential equation without any "extra" functions added to the system. Mathematically, it appears as an equation where every term is a function or derivative of the unknown function. The homogeneous part of the equation y+λ2y=0 disregards any external forces or inhomogeneous terms. This leads us to work with natural behaviors produced by the system itself.

Homogeneous equations allow us to understand and solve for inherent characteristics of the system, like oscillations or decay patterns, that come from the properties of the differential operator itself. By concentrating solely on the homogeneous part, we position ourselves to derive solutions using characteristic equations, yielding general solutions comprised of exponentials or trigonometric functions depending on the roots of the characteristic equation.
Particular Solution
The particular solution specifically addresses the inhomogeneous equation components. In our problem, the inhomogeneity comes from m=1Namsinmπt, implying periodic external forces acting on the system. Deriving the particular solution involves seeking a function that satisfies the complete differential equation as given, including the non-homogeneity.

To find the particular solution for a term like amsinmπt, assume a potential solution form such as yp(t)=Cmsinmπt. Substituting back into the differential equation and matching coefficients helps in determining Cm. We deduce:
  • Cm=amλ2m2π2
This ensures that the proposed form aligns perfectly with the inhomogeneous part of the equation. The complete particular solution is then:
  • yp(t)=m=1Namλ2m2π2sinmπt
This step-by-step approach provides individual terms that we eventually sum up to represent the total external force response.
Inhomogeneous Equation
Inhomogeneous equations are differential equations that include some external force or bias in addition to the regular terms of the equation. The inhomogeneity typically appears as an added function, such as the sum of sine functions m=1Namsinmπt found in this differential equation problem.

Dealing with inhomogeneous equations involves accounting not only for the inherent system properties but also for how these external factors disturb or alter the system's behavior. The strategy is to decompose the general solution into the complementary solution, which handles the homogeneous part, and the particular solution, capturing the effects of the added non-homogeneous function. Therefore, a general solution appears as follows:
  • y(t)=Acos(λt)+Bsin(λt)+m=1Namλ2m2π2sinmπt
This solution structure effectively integrates both natural system responses and external influences, offering a complete picture of the system dynamics.

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