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Consider a general first order linear non-homogeneous ordinary differential equation with y = φ(x) as its solution.
Let ψ(x) be the solution of the associated homogeneous equation and C an arbitrary constant. Then, the general equation has the following solution.
Let ψ(x) be the solution of the associated homogeneous equation and C an arbitrary constant. Then, the general equation has the following solution.
- a)y = Cφ(x)
- b)y = Cφ(x) + ψ(x)
- c)y = Cψ(x) + φ(x)
- d)y = Cψ(x)
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Consider a general first order linear non-homogeneous ordinary differe...
Y(x) = y0(x) + yp(x)
where y0(x) is the general solution to the corresponding homogeneous equation and yp(x) is a particular solution to the non-homogeneous equation.
To find yp(x), we can use the method of undetermined coefficients or variation of parameters.
1. Method of undetermined coefficients:
In this method, we assume a form for yp(x) based on the non-homogeneous term in the equation and solve for the coefficients. The form of yp(x) depends on the type of non-homogeneous term.
For example, if the non-homogeneous term is a constant term, yp(x) can be assumed to be a constant. If the non-homogeneous term is a polynomial of degree n, yp(x) can be assumed to be a polynomial of degree n with undetermined coefficients. If the non-homogeneous term is an exponential function, yp(x) can be assumed to be an exponential function with undetermined coefficients, and so on.
By substituting the assumed form of yp(x) into the non-homogeneous equation, we can determine the values of the undetermined coefficients and obtain a particular solution yp(x).
2. Variation of parameters:
In this method, we assume the particular solution yp(x) to have the form yp(x) = u(x)y0(x), where u(x) is a function to be determined.
Substituting this form into the non-homogeneous equation, we can obtain an equation for u(x). Solving this equation, we can find u(x) and hence obtain yp(x).
Once we have both y0(x) and yp(x), the general solution to the non-homogeneous equation is given by y(x) = y0(x) + yp(x).
Note that for a first order linear non-homogeneous ordinary differential equation, the general solution can also be expressed using an integrating factor method, which involves multiplying the entire equation by an integrating factor to make it exact. However, this method is not as commonly used as the methods mentioned above.
where y0(x) is the general solution to the corresponding homogeneous equation and yp(x) is a particular solution to the non-homogeneous equation.
To find yp(x), we can use the method of undetermined coefficients or variation of parameters.
1. Method of undetermined coefficients:
In this method, we assume a form for yp(x) based on the non-homogeneous term in the equation and solve for the coefficients. The form of yp(x) depends on the type of non-homogeneous term.
For example, if the non-homogeneous term is a constant term, yp(x) can be assumed to be a constant. If the non-homogeneous term is a polynomial of degree n, yp(x) can be assumed to be a polynomial of degree n with undetermined coefficients. If the non-homogeneous term is an exponential function, yp(x) can be assumed to be an exponential function with undetermined coefficients, and so on.
By substituting the assumed form of yp(x) into the non-homogeneous equation, we can determine the values of the undetermined coefficients and obtain a particular solution yp(x).
2. Variation of parameters:
In this method, we assume the particular solution yp(x) to have the form yp(x) = u(x)y0(x), where u(x) is a function to be determined.
Substituting this form into the non-homogeneous equation, we can obtain an equation for u(x). Solving this equation, we can find u(x) and hence obtain yp(x).
Once we have both y0(x) and yp(x), the general solution to the non-homogeneous equation is given by y(x) = y0(x) + yp(x).
Note that for a first order linear non-homogeneous ordinary differential equation, the general solution can also be expressed using an integrating factor method, which involves multiplying the entire equation by an integrating factor to make it exact. However, this method is not as commonly used as the methods mentioned above.
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Consider a general first order linear non-homogeneous ordinary differential equation with y = φ(x) as its solution.Let ψ(x) be the solution of the associated homogeneous equation and C an arbitrary constant. Then, the general equation has the following solution.a)y = Cφ(x)b)y = Cφ(x) +ψ(x)c)y = Cψ(x) +φ(x)d)y = Cψ(x)Correct answer is option 'C'. Can you explain this answer?
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Consider a general first order linear non-homogeneous ordinary differential equation with y = φ(x) as its solution.Let ψ(x) be the solution of the associated homogeneous equation and C an arbitrary constant. Then, the general equation has the following solution.a)y = Cφ(x)b)y = Cφ(x) +ψ(x)c)y = Cψ(x) +φ(x)d)y = Cψ(x)Correct answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Consider a general first order linear non-homogeneous ordinary differential equation with y = φ(x) as its solution.Let ψ(x) be the solution of the associated homogeneous equation and C an arbitrary constant. Then, the general equation has the following solution.a)y = Cφ(x)b)y = Cφ(x) +ψ(x)c)y = Cψ(x) +φ(x)d)y = Cψ(x)Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a general first order linear non-homogeneous ordinary differential equation with y = φ(x) as its solution.Let ψ(x) be the solution of the associated homogeneous equation and C an arbitrary constant. Then, the general equation has the following solution.a)y = Cφ(x)b)y = Cφ(x) +ψ(x)c)y = Cψ(x) +φ(x)d)y = Cψ(x)Correct answer is option 'C'. Can you explain this answer?.
Consider a general first order linear non-homogeneous ordinary differential equation with y = φ(x) as its solution.Let ψ(x) be the solution of the associated homogeneous equation and C an arbitrary constant. Then, the general equation has the following solution.a)y = Cφ(x)b)y = Cφ(x) +ψ(x)c)y = Cψ(x) +φ(x)d)y = Cψ(x)Correct answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Consider a general first order linear non-homogeneous ordinary differential equation with y = φ(x) as its solution.Let ψ(x) be the solution of the associated homogeneous equation and C an arbitrary constant. Then, the general equation has the following solution.a)y = Cφ(x)b)y = Cφ(x) +ψ(x)c)y = Cψ(x) +φ(x)d)y = Cψ(x)Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a general first order linear non-homogeneous ordinary differential equation with y = φ(x) as its solution.Let ψ(x) be the solution of the associated homogeneous equation and C an arbitrary constant. Then, the general equation has the following solution.a)y = Cφ(x)b)y = Cφ(x) +ψ(x)c)y = Cψ(x) +φ(x)d)y = Cψ(x)Correct answer is option 'C'. Can you explain this answer?.
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