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Chapter 4: Problem 110

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime}(x)=1, F^{\prime}(0)=3, F(0)=4$$

Short Answer

Expert verified
Question: Determine the function F(x) that satisfies the given second-order differential equation \(F^{\prime \prime}(x) = 1\) with the initial conditions \(F^{\prime}(0) = 3\) and \(F(0) = 4\). Answer: The function F(x) that satisfies the given differential equations and initial conditions is \(F(x) = \frac{1}{2}x^2 + 3x + 4\).

Step by step solution

01

Integrate the second-order differential equation once

Integrate the given differential equation, \(F^{\prime \prime}(x)=1\), with respect to x. $$\int F^{\prime \prime}(x) dx = \int 1 dx$$ Integration results in: $$F^{\prime}(x) = x + C_1$$ Notice we have a constant of integration, \(C_1\), which we will determine using the given initial condition \(F^{\prime}(0)=3\).
02

Use initial condition to find \(C_1\)

\(F^{\prime}(0) = 3\). Substitute the initial condition into the equation we found in step 1: $$F^{\prime}(0) = 0 + C_1 = 3$$ So, the constant of integration, \(C_1=3\). Our first-order differential equation is now: $$F^{\prime}(x) = x + 3$$
03

Integrate the first-order differential equation

Integrate the first-order differential equation, \(F^{\prime}(x) = x + 3\), with respect to x. $$\int F^{\prime}(x) dx = \int (x + 3) dx$$ Integration results in: $$F(x) = \frac{1}{2}x^2 + 3x + C_2$$ Notice that we have another constant of integration, \(C_2\), which we will determine using the given initial condition \(F(0)=4\).
04

Use initial condition to find \(C_2\)

\(F(0) = 4\). Substitute the initial condition into the equation we found in step 3: $$F(0) = \frac{1}{2}(0)^2 + 3(0) + C_2 = 4$$ So, the constant of integration, \(C_2=4\).
05

Write the final function

Substitute the constants of integration \(C_1\) and \(C_2\) we found in the previous steps into the function: $$F(x) = \frac{1}{2}x^2 + 3x + 4$$ The function F(x) that satisfies the given differential equations and initial conditions is: $$F(x) = \frac{1}{2}x^2 + 3x + 4$$

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