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(minus​1)equalsnegative 8.

Solution

To find the function ( y(x) ) satisfying the given differential equation dydx=3x67\frac{dy}{dx} = \frac{3x^{-6}}{7} and the initial condition ( y(-1) = -8 ), we need to perform the following steps:

  1. Solve the differential equation:

    dydx=37x6\frac{dy}{dx} = \frac{3}{7} x^{-6}

    Integrate both sides with respect to xx:

    y=37x6dxy = \int \frac{3}{7} x^{-6} \, dx

    y=37x6dxy = \frac{3}{7} \int x^{-6} \, dx

    Recall that the integral of xnx^n is xn+1n+1\frac{x^{n+1}}{n+1} (for n1n \neq -1):

    y=37x55y = \frac{3}{7} \cdot \frac{x^{-5}}{-5}

    Simplify the constant factor:

    y=335x5+Cy = -\frac{3}{35} x^{-5} + C

    So the general solution is:

    y=335x5+Cy = -\frac{3}{35} x^{-5} + C

  2. Apply the initial condition ( y(-1) = -8 ):

    8=335(1)5+C-8 = -\frac{3}{35} (-1)^{-5} + C

    Since ( (-1)^{-5} = -1 ):

    8=335(1)+C-8 = -\frac{3}{35} (-1) + C

    8=335+C-8 = \frac{3}{35} + C

    Solve for CC:

    C=8335C = -8 - \frac{3}{35}

    Convert 8-8 to a fraction with a common denominator of 35:

    8=28035-8 = -\frac{280}{35}

    So:

    C=28035335=28335C = -\frac{280}{35} - \frac{3}{35} = -\frac{283}{35}

  3. Write the particular solution:

    y=335x528335y = -\frac{3}{35} x^{-5} - \frac{283}{35}

    Simplify the final function if possible:

    y=335x528335y = -\frac{3}{35} x^{-5} - \frac{283}{35}

Therefore, the function ( y(x) ) that satisfies the given differential equation and initial condition is:

y=335x528335y = -\frac{3}{35} x^{-5} - \frac{283}{35}

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:

  1. How do you solve a first-order differential equation?
  2. What are initial conditions in differential equations?
  3. How do you integrate power functions?
  4. What is the importance of integrating factors in differential equations?
  5. 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