/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 113 Find the function \(F\) that sat... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 113

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime \prime}(x)=672 x^{5}+24 x, F^{\prime \prime}(0)=0, F^{\prime}(0)=2, F(0)=1$$

Short Answer

Expert verified
Answer: The function F(x) that satisfies the given differential equations and initial conditions is: \(F(x) = 2x^8 + x^4 + 2x + 1\).

Step by step solution

01

Integrate the Third Derivative

Firstly, integrate the given third derivative \( F^{\prime \prime \prime}(x) = 672x^5 + 24x \) with respect to \(x\) to find the second derivative, \(F^{\prime \prime}(x)\). $$\int(672x^5 + 24x) dx = 672\int(x^5) dx + 24\int (x) dx = 672\frac{x^6}{6} + 24\frac{x^2}{2} + C_1$$ $$F^{\prime \prime}(x) = 112x^6 + 12x^2 + C_1$$ Now use the initial condition \(F^{\prime \prime}(0) = 0\) to find the constant \(C_1\). $$F^{\prime \prime}(0) = 112(0)^6 + 12(0)^2 + C_1 = C_1$$ Since \(F^{\prime \prime}(0) = 0\), we get \(C_1=0\). Thus, $$F^{\prime \prime}(x) = 112x^6 + 12x^2$$
02

Integrate the Second Derivative

Next, integrate the obtained second derivative, \(F^{\prime \prime}(x) = 112x^6 + 12x^2\), with respect to \(x\) to find the first derivative, \(F^{\prime}(x)\). $$\int(112x^6 + 12x^2) dx = 112\int(x^6) dx + 12\int (x^2) dx = 112\frac{x^7}{7} + 12\frac{x^3}{3} + C_2$$ $$F^{\prime}(x) = 16x^7 + 4x^3 + C_2$$ Now, use the initial condition \(F^{\prime}(0) = 2\) to find the constant \(C_2\). $$F^{\prime}(0) = 16(0)^7 + 4(0)^3 + C_2 = C_2$$ Since \(F^{\prime}(0) = 2\), we get \(C_2=2\). Thus, $$F^{\prime}(x) = 16x^7 + 4x^3 + 2$$
03

Integrate the First Derivative

Finally, integrate the obtained first derivative, \(F^{\prime}(x) = 16x^7 + 4x^3 + 2\), with respect to \(x\) to find the function F(x). $$\int(16x^7 + 4x^3 + 2) dx = 16\int(x^7) dx + 4\int (x^3) dx + 2\int(1)dx = 16\frac{x^8}{8} + 4\frac{x^4}{4} + 2x + C_3$$ $$F(x) = 2x^8 + x^4 + 2x + C_3$$ Now, use the initial condition \(F(0) = 1\) to find the constant \(C_3\). $$F(0) = 2(0)^8 + (0)^4 + 2(0) + C_3 = C_3$$ Since \(F(0) = 1\), we get \(C_3=1\). Thus, $$F(x) = 2x^8 + x^4 + 2x + 1$$ The function \(F\) that satisfies the given differential equations and initial conditions is: $$F(x) = 2x^8 + x^4 + 2x + 1$$

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