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Chapter 4: Problem 111

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

Short Answer

Expert verified
**Question:** Determine the function \(F(x)\) that satisfies the given second-order differential equation and initial conditions: \(F^{\prime\prime}(x) = \cos(x)\) \(F(0) = 4\) \(F^{\prime}(0) = 3\) **Answer:** The function \(F(x)\) that satisfies the given differential equation and initial conditions is: $$F(x) = -\cos(x) + 3x + 3 - 3\pi$$

Step by step solution

01

Integrate first-order differential equation

Integrate the given differential equation, \(F^{\prime\prime}(x)=\cos(x)\), with respect to \(x\) to get \(F'(x)\). $$F'(x) = \int \cos(x) dx$$ Recall that the integral of \(\cos(x)\) is \(\sin(x)\). However, there's one constant of integration, \(C_1\): $$F'(x) = \sin(x) + C_1$$
02

Apply first initial condition

Now, use the given initial condition \(F^{\prime}(0) = 3\) to find \(C_1\). Plug in \(x = 0\) into the equation for \(F'(x)\): $$3 = \sin(0) + C_1$$ Since the \(\sin(0) = 0\), we have: $$C_1 = 3$$ So, the first derivative \(F'(x)\) is given by: $$F'(x) = \sin(x) + 3$$
03

Integrate to find the original function

Now, we'll integrate \(F'(x)\) to find the original function \(F(x)\): $$F(x) = \int (\sin(x) + 3) dx$$ Integrating \(\sin(x)\) gives \(-\cos(x)\) and integrating \(3\) gives \(3x\). But, there's one more constant of integration, \(C_2\): $$F(x) = -\cos(x) + 3x + C_2$$
04

Apply second initial condition

Now, use the given initial condition \(F(\pi) = 4\) to find \(C_2\). Plug in \(x=\pi\) into the equation for \(F(x)\): $$4 = -\cos(\pi) + 3\pi + C_2$$ Recall that \(\cos(\pi) = -1\), so the equation becomes: $$4 = -(-1) + 3\pi + C_2$$ Solve for \(C_2\): $$C_2 = 4 - 1 - 3\pi$$ $$C_2 = 3 - 3\pi$$
05

Write the final function

Finally, we found the constants \(C_1\) and \(C_2\). Now, we can write down the final function \(F(x)\): $$F(x) = -\cos(x) + 3x + (3 - 3\pi)$$ The function \(F(x)\) that satisfies the given differential equation and initial conditions is: $$F(x) = -\cos(x) + 3x + 3 - 3\pi$$

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