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Magazine Chapter 4: Problem 50
In Problems \(37-52\) solve the given differential equation subject to the indicated initial conditions. \(\frac{d^{4} y}{d x^{4}}=0, \quad y(0)=2, y^{\prime}(0)=3, y^{\prime \prime}(0)=4, y^{\prime \prime \prime}(0)=5\)
Short Answer
Expert verified
The solution is \( y = \frac{5}{6} x^3 + 2x^2 + 3x + 2 \).
Step by step solution
01
Integrate the Differential Equation
To begin solving the fourth-order differential equation \( \frac{d^4 y}{dx^4} = 0 \), we first integrate it four times with respect to \( x \) to find the expression for \( y \). The first integration gives \( \frac{d^3 y}{dx^3} = C_1 \).
02
Second Integration
Integrate \( \frac{d^3 y}{dx^3} = C_1 \) with respect to \( x \) to get \( \frac{d^2 y}{dx^2} = C_1 x + C_2 \).
03
Third Integration
Integrate \( \frac{d^2 y}{dx^2} = C_1 x + C_2 \) with respect to \( x \) to obtain \( \frac{d y}{dx} = \frac{1}{2} C_1 x^2 + C_2 x + C_3 \).
04
Fourth Integration
Integrate \( \frac{d y}{dx} = \frac{1}{2} C_1 x^2 + C_2 x + C_3 \) with respect to \( x \) to find \( y = \frac{1}{6} C_1 x^3 + \frac{1}{2} C_2 x^2 + C_3 x + C_4 \).
05
Apply Initial Conditions
Apply the initial conditions to determine the constants. Start with \( y(0) = 2 \), which gives \( C_4 = 2 \).
06
Use First Derivative Condition
Use the condition \( y'(0) = 3 \). Substituting into \( \frac{d y}{dx} = \frac{1}{2} C_1 x^2 + C_2 x + C_3 \), we find \( C_3 = 3 \).
07
Use Second Derivative Condition
Apply \( y''(0) = 4 \) in \( \frac{d^2 y}{dx^2} = C_1 x + C_2 \), yielding \( C_2 = 4 \).
08
Use Third Derivative Condition
Finally, apply \( y'''(0) = 5 \) in \( \frac{d^3 y}{dx^3} = C_1 \), which gives \( C_1 = 5 \).
09
Write Final Solution
Substitute all constants back into the expression for \( y \): \[ y = \frac{5}{6} x^3 + 2 x^2 + 3x + 2 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fourth-order differential equations
A fourth-order differential equation involves derivatives up to the fourth degree with respect to an independent variable, typically denoted as \( x \). The given equation \( \frac{d^4 y}{dx^4} = 0 \) is a simple example. Fourth-order equations appear in various fields like engineering and physics, often describing systems with higher-order dynamics. These equations can be more complex and require multiple integrations to solve.
- Fourth-order means four derivatives need to be taken.
- In this specific problem, the fourth derivative being zero simplifies the integration process.
Integration of differential equations
Integrating a differential equation is key to finding a solution that describes the behavior of a system. This process involves reversing differentiation to find the original function from its derivatives. With each integration step, a constant of integration (denoted \( C \)) appears.
In the problem, we integrate the function four times:
In the problem, we integrate the function four times:
- The first integration turns \( \frac{d^4 y}{dx^4} = 0 \) into \( \frac{d^3 y}{dx^3} = C_1 \).
- Further integration results in \( \frac{d^2 y}{dx^2} = C_1 x + C_2 \).
- Continued integration leads to \( \frac{d y}{dx} = \frac{1}{2} C_1 x^2 + C_2 x + C_3 \).
- The final integration provides \( y = \frac{1}{6} C_1 x^3 + \frac{1}{2} C_2 x^2 + C_3 x + C_4 \).
Initial conditions
Initial conditions are crucial in solving differential equations, particularly when multiple integrations introduce constants. These conditions often describe the state of a system at a specific point, allowing us to determine the constants and thus specify a particular solution.
In the exercise, several initial conditions were provided:
In the exercise, several initial conditions were provided:
- \( y(0) = 2 \), which sets \( C_4 = 2 \). This corresponds to the value of \( y \) when \( x=0 \).
- \( y'(0) = 3 \), which gives \( C_3 = 3 \) by substituting into the first derivative equation.
- \( y''(0) = 4 \), determining \( C_2 = 4 \) through substitution.
- \( y'''(0) = 5 \), finding \( C_1 = 5 \) with the third derivative equation.
Constant of integration
The constants of integration, \( C_1, C_2, C_3, \) and \( C_4 \), arise during the integration process. These constants are undetermined during integration because integration is the inverse of differentiation, which ignores constant terms.
- In a typical differential equation, each integration introduces its own constant.
- The constants must be determined using initial or boundary conditions.
