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Let g:R-R be a continuous function.Which one of the following is the solution of the differential equations
d^2y/dx^2+y=g(x) for x€R satisfied the continuous y(0)=0,y'(0)=1?
d^2y/dx^2+y=g(x) for x€R satisfied the continuous y(0)=0,y'(0)=1?
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Let g:R-R be a continuous function.Which one of the following is the s...
Understanding the Differential Equation
The given differential equation is:
- d²y/dx² + y = g(x)
This is a second-order linear ordinary differential equation with a non-homogeneous term g(x).
General Solution Structure
The solution to this equation can be expressed as:
- y(x) = y_h(x) + y_p(x)
Where:
- y_h(x) is the general solution of the homogeneous equation (d²y/dx² + y = 0).
- y_p(x) is a particular solution to the non-homogeneous equation.
Finding the Homogeneous Solution
The homogeneous part can be solved by assuming a solution of the form e^(rx), leading to the characteristic equation:
- r² + 1 = 0
This gives complex roots r = i and r = -i, leading to:
- y_h(x) = A cos(x) + B sin(x)
Where A and B are constants determined by initial conditions.
Finding the Particular Solution
For y_p(x), methods like undetermined coefficients or variation of parameters can be employed based on the form of g(x).
Applying Initial Conditions
The initial conditions provided are:
- y(0) = 0
- y'(0) = 1
Using these, we can determine the constants A and B.
1. From y(0) = 0:
- A cos(0) + B sin(0) + y_p(0) = 0
- This simplifies to A + y_p(0) = 0.
2. From y'(0) = 1:
- -A sin(0) + B cos(0) + y_p'(0) = 1
- This simplifies to B + y_p'(0) = 1.
Conclusion
By solving for A, B, and incorporating the particular solution, you obtain the complete solution y(x) that satisfies the differential equation as well as the initial conditions. Note that the specific form of g(x) will affect y_p(x) and hence the final solution.
The given differential equation is:
- d²y/dx² + y = g(x)
This is a second-order linear ordinary differential equation with a non-homogeneous term g(x).
General Solution Structure
The solution to this equation can be expressed as:
- y(x) = y_h(x) + y_p(x)
Where:
- y_h(x) is the general solution of the homogeneous equation (d²y/dx² + y = 0).
- y_p(x) is a particular solution to the non-homogeneous equation.
Finding the Homogeneous Solution
The homogeneous part can be solved by assuming a solution of the form e^(rx), leading to the characteristic equation:
- r² + 1 = 0
This gives complex roots r = i and r = -i, leading to:
- y_h(x) = A cos(x) + B sin(x)
Where A and B are constants determined by initial conditions.
Finding the Particular Solution
For y_p(x), methods like undetermined coefficients or variation of parameters can be employed based on the form of g(x).
Applying Initial Conditions
The initial conditions provided are:
- y(0) = 0
- y'(0) = 1
Using these, we can determine the constants A and B.
1. From y(0) = 0:
- A cos(0) + B sin(0) + y_p(0) = 0
- This simplifies to A + y_p(0) = 0.
2. From y'(0) = 1:
- -A sin(0) + B cos(0) + y_p'(0) = 1
- This simplifies to B + y_p'(0) = 1.
Conclusion
By solving for A, B, and incorporating the particular solution, you obtain the complete solution y(x) that satisfies the differential equation as well as the initial conditions. Note that the specific form of g(x) will affect y_p(x) and hence the final solution.
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Let g:R-R be a continuous function.Which one of the following is the solution of the differential equations d^2y/dx^2+y=g(x) for x€R satisfied the continuous y(0)=0,y'(0)=1?
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Let g:R-R be a continuous function.Which one of the following is the solution of the differential equations d^2y/dx^2+y=g(x) for x€R satisfied the continuous y(0)=0,y'(0)=1? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let g:R-R be a continuous function.Which one of the following is the solution of the differential equations d^2y/dx^2+y=g(x) for x€R satisfied the continuous y(0)=0,y'(0)=1? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let g:R-R be a continuous function.Which one of the following is the solution of the differential equations d^2y/dx^2+y=g(x) for x€R satisfied the continuous y(0)=0,y'(0)=1?.
Let g:R-R be a continuous function.Which one of the following is the solution of the differential equations d^2y/dx^2+y=g(x) for x€R satisfied the continuous y(0)=0,y'(0)=1? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let g:R-R be a continuous function.Which one of the following is the solution of the differential equations d^2y/dx^2+y=g(x) for x€R satisfied the continuous y(0)=0,y'(0)=1? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let g:R-R be a continuous function.Which one of the following is the solution of the differential equations d^2y/dx^2+y=g(x) for x€R satisfied the continuous y(0)=0,y'(0)=1?.
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