Math Problem Statement
Find the function y(x) satisfying StartFraction dy Over dx EndFraction equals 3 x Superscript negative 6 divided by 7 and y(minus1)equalsnegative 8.
Solution
To find the function ( y(x) ) satisfying the given differential equation and the initial condition ( y(-1) = -8 ), we need to perform the following steps:
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Solve the differential equation:
Integrate both sides with respect to :
Recall that the integral of is (for ):
Simplify the constant factor:
So the general solution is:
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Apply the initial condition ( y(-1) = -8 ):
Since ( (-1)^{-5} = -1 ):
Solve for :
Convert to a fraction with a common denominator of 35:
So:
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Write the particular solution:
Simplify the final function if possible:
Therefore, the function ( y(x) ) that satisfies the given differential equation and initial condition is:
Would you like more details on any of these steps or have any other questions? Here are some related questions you might be interested in:
- How do you solve a first-order differential equation?
- What are initial conditions in differential equations?
- How do you integrate power functions?
- What is the importance of integrating factors in differential equations?
- How do you handle differential equations with multiple initial conditions?
Tip: Always check your work by substituting the initial condition back into the solution to ensure consistency.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Integration
Formulas
Power rule for integration
Theorems
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Suitable Grade Level
College level
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