Question 32 of 96intermediate🔧 ApplyLong Answer4 marks

For each of the differential equations in Exercises 1 to 10, find the general solution:

x5dydx=−y5x^5 \frac{dy}{dx} = -y^5
Correct Answer

The given differential equation is:

x5dydx=−y5x^5 \frac{dy}{dx} = -y^5

Separating the variables xx and yy, we get:

dyy5=−dxx5\frac{dy}{y^5} = -\frac{dx}{x^5}

Integrating both sides of the equation:

∫1y5 dy=−∫1x5 dx\int \frac{1}{y^5} \, dy = - \int \frac{1}{x^5} \, dx ∫y−5 dy=−∫x−5 dx\int y^{-5} \, dy = - \int x^{-5} \, dx

Using the standard integral formula ∫xndx=xn+1n+1\int x^n dx = \frac{x^{n+1}}{n+1}, we get:

[y−5+1−5+1]=−[x−5+1−5+1]+C\left[ \frac{y^{-5+1}}{-5+1} \right] = - \left[ \frac{x^{-5+1}}{-5+1} \right] + C y−4−4=−x−4−4+C\frac{y^{-4}}{-4} = - \frac{x^{-4}}{-4} + C

Simplifying the equation:

−14y4=14x4+C-\frac{1}{4y^4} = \frac{1}{4x^4} + C

Multiplying the entire equation by −4-4 and rearranging the terms, the general solution is:

1x4+1y4=C1\frac{1}{x^4} + \frac{1}{y^4} = C_1

(where C1C_1 is an arbitrary constant).

Exercise: EXERCISE 9.3 | Q: 8 | (Page 12)
For More Understanding

Explanation

This question involves finding the general solution for a differential equation where the variables are separable. Following the method illustrated in Example 5 and Example 6 of the provided context, the first step is to rearrange the equation to isolate terms containing yy (with dydy) on one side and terms containing xx (with dxdx) on the other.

Once separated, both sides are integrated individually. The integration follows the power rule, resulting in negative powers since the original terms were in the denominator.

Finally, the constants of integration are combined into a single arbitrary constant to present the general solution clearly.

Solution Steps

  1. Step 1: Rewrite the given differential equation x5dydx=−y5x^5 \frac{dy}{dx} = -y^5 to separate variables:

    dyy5=−dxx5\frac{dy}{y^5} = -\frac{dx}{x^5}

  2. Step 2: Integrate both sides with respect to their respective variables:

    ∫y−5 dy=−∫x−5 dx\int y^{-5} \, dy = - \int x^{-5} \, dx

  3. Step 3: Apply the power rule for integration (∫xndx=xn+1n+1\int x^n dx = \frac{x^{n+1}}{n+1}) to get:

    y−4−4=−x−4−4+C\frac{y^{-4}}{-4} = -\frac{x^{-4}}{-4} + C

  4. Step 4: Simplify the equation to:

    −14y4=14x4+C-\frac{1}{4y^4} = \frac{1}{4x^4} + C

  5. Step 5: Multiply by −4-4 and rearrange to find the general solution:

    1x4+1y4=C1\frac{1}{x^4} + \frac{1}{y^4} = C_1
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