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Chapter 4: Problem 76

Find the solution of the following initial value problems. $$v^{\prime}(x)=4 x^{1 / 3}+2 x^{-1 / 3} ; v(8)=40$$

Short Answer

Expert verified
Question: Find the particular solution for the initial value problem $$v^{\prime}(x)=4x^{\frac{1}{3}}+2x^{-\frac{1}{3}}$$ given the initial condition $$v(8) = 40$$. Answer: The particular solution to the given initial value problem is $$v(x) = 3x^{\frac{4}{3}} + 3x^{\frac{2}{3}} - 18.289$$.

Step by step solution

01

Identify the differential equation

The given differential equation is $$v^{\prime}(x)=4x^{\frac{1}{3}}+2x^{-\frac{1}{3}}$$ which is a first-order differential equation.
02

Integrate both sides

To find the general solution of the first-order differential equation, we need to integrate both sides: $$\int v^{\prime}(x) dx = \int (4x^{\frac{1}{3}}+2x^{-\frac{1}{3}}) dx$$
03

Solve the integral

Now, we will integrate the terms individually. $$\int 4x^{\frac{1}{3}} dx + \int 2x^{-\frac{1}{3}} dx$$ Using the power rule for integration, $$\int x^n dx =\frac{x^{n+1}}{n+1} + C$$ we get, $$\frac{3}{4} (2x^{\frac{4}{3}}) + \frac{3}{2} (x^{\frac{2}{3}}) + C$$ which simplifies to $$3x^{\frac{4}{3}} + 3x^{\frac{2}{3}} + C.$$ Now, the general solution for the given differential equation is: $$v(x) = 3x^{\frac{4}{3}} + 3x^{\frac{2}{3}} + C$$
04

Use initial condition to find the particular solution

To find the particular solution, we'll use the given initial condition $$v(8) = 40$$. Substitute the value of x with 8 in the function v(x): $$v(8) = 3(8)^{\frac{4}{3}} + 3(8)^{\frac{2}{3}} + C$$ Now, calculate the value of C: $$40 = 3(8)^{\frac{4}{3}} + 3(8)^{\frac{2}{3}} + C$$ $$C = 40 - 3(8)^{\frac{4}{3}} - 3(8)^{\frac{2}{3}}$$ $$C \approx -18.289$$
05

Write down the particular solution

Now that we have found the value of C, we can write the particular solution to the given initial value problem: $$v(x) = 3x^{\frac{4}{3}} + 3x^{\frac{2}{3}} - 18.289$$

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