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Let y"+ p(x) y' + q(x) y = 0, x belongs to R, where p(x) and q(x) are continuous function if y₁ = 10sin3x - 12cos and y₂ = 15sin3x 17cos3x are two linearly independent solution of differential equation then 10p(10)+ 15g(15) is?
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Let y"+ p(x) y' + q(x) y = 0, x belongs to R, where p(x) and q(x) are ...
Given:
We are given a second-order linear homogeneous differential equation of the form y" + p(x)y' + q(x)y = 0, where p(x) and q(x) are continuous functions.
We are also given two linearly independent solutions of the differential equation:
y₁ = 10sin3x - 12cos3x
y₂ = 15sin3x + 17cos3x
To find:
We need to find the values of 10p(10) and 15q(15), where p(x) and q(x) are the coefficients of y' and y in the differential equation, respectively.
Solution:
To find the values of 10p(10) and 15q(15), we need to determine the functions p(x) and q(x) first.
Step 1: Find the first derivative of y₁:
y₁ = 10sin3x - 12cos3x
Differentiating both sides with respect to x:
y₁' = 10(3cos3x) + 12(3sin3x) = 30cos3x + 36sin3x
Step 2: Find the second derivative of y₁:
Differentiating y₁' with respect to x:
y₁" = -30(3sin3x) + 36(3cos3x) = -90sin3x + 108cos3x
Step 3: Substitute y₁, y₁', and y₁" into the differential equation:
y" + p(x)y' + q(x)y = 0
(-90sin3x + 108cos3x) + p(x)(30cos3x + 36sin3x) + q(x)(10sin3x - 12cos3x) = 0
Step 4: Simplify the equation:
Matching the coefficients of sin3x and cos3x terms separately, we get:
-90 + 30p(x) + 10q(x) = 0
108 + 36p(x) - 12q(x) = 0
Step 5: Solve the system of equations:
Solving the system of equations, we get:
p(x) = 6
q(x) = 9
Step 6: Find the values of 10p(10) and 15q(15):
Now, we can substitute the values of p(x) and q(x) into the expressions and evaluate them:
10p(10) = 10(6) = 60
15g(15) = 15(9) = 135
We are given a second-order linear homogeneous differential equation of the form y" + p(x)y' + q(x)y = 0, where p(x) and q(x) are continuous functions.
We are also given two linearly independent solutions of the differential equation:
y₁ = 10sin3x - 12cos3x
y₂ = 15sin3x + 17cos3x
To find:
We need to find the values of 10p(10) and 15q(15), where p(x) and q(x) are the coefficients of y' and y in the differential equation, respectively.
Solution:
To find the values of 10p(10) and 15q(15), we need to determine the functions p(x) and q(x) first.
Step 1: Find the first derivative of y₁:
y₁ = 10sin3x - 12cos3x
Differentiating both sides with respect to x:
y₁' = 10(3cos3x) + 12(3sin3x) = 30cos3x + 36sin3x
Step 2: Find the second derivative of y₁:
Differentiating y₁' with respect to x:
y₁" = -30(3sin3x) + 36(3cos3x) = -90sin3x + 108cos3x
Step 3: Substitute y₁, y₁', and y₁" into the differential equation:
y" + p(x)y' + q(x)y = 0
(-90sin3x + 108cos3x) + p(x)(30cos3x + 36sin3x) + q(x)(10sin3x - 12cos3x) = 0
Step 4: Simplify the equation:
Matching the coefficients of sin3x and cos3x terms separately, we get:
-90 + 30p(x) + 10q(x) = 0
108 + 36p(x) - 12q(x) = 0
Step 5: Solve the system of equations:
Solving the system of equations, we get:
p(x) = 6
q(x) = 9
Step 6: Find the values of 10p(10) and 15q(15):
Now, we can substitute the values of p(x) and q(x) into the expressions and evaluate them:
10p(10) = 10(6) = 60
15g(15) = 15(9) = 135
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Let y"+ p(x) y' + q(x) y = 0, x belongs to R, where p(x) and q(x) are continuous function if y₁ = 10sin3x - 12cos and y₂ = 15sin3x 17cos3x are two linearly independent solution of differential equation then 10p(10)+ 15g(15) is?
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Let y"+ p(x) y' + q(x) y = 0, x belongs to R, where p(x) and q(x) are continuous function if y₁ = 10sin3x - 12cos and y₂ = 15sin3x 17cos3x are two linearly independent solution of differential equation then 10p(10)+ 15g(15) is? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let y"+ p(x) y' + q(x) y = 0, x belongs to R, where p(x) and q(x) are continuous function if y₁ = 10sin3x - 12cos and y₂ = 15sin3x 17cos3x are two linearly independent solution of differential equation then 10p(10)+ 15g(15) is? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let y"+ p(x) y' + q(x) y = 0, x belongs to R, where p(x) and q(x) are continuous function if y₁ = 10sin3x - 12cos and y₂ = 15sin3x 17cos3x are two linearly independent solution of differential equation then 10p(10)+ 15g(15) is?.
Let y"+ p(x) y' + q(x) y = 0, x belongs to R, where p(x) and q(x) are continuous function if y₁ = 10sin3x - 12cos and y₂ = 15sin3x 17cos3x are two linearly independent solution of differential equation then 10p(10)+ 15g(15) is? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let y"+ p(x) y' + q(x) y = 0, x belongs to R, where p(x) and q(x) are continuous function if y₁ = 10sin3x - 12cos and y₂ = 15sin3x 17cos3x are two linearly independent solution of differential equation then 10p(10)+ 15g(15) is? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let y"+ p(x) y' + q(x) y = 0, x belongs to R, where p(x) and q(x) are continuous function if y₁ = 10sin3x - 12cos and y₂ = 15sin3x 17cos3x are two linearly independent solution of differential equation then 10p(10)+ 15g(15) is?.
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