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Chapter 1: Problem 11

Verify that the indicated function is a solution of the given differential equation. $$ y^{\prime \prime \prime}-2 y^{3}-y^{\prime}+2 y=6 ; \quad y=c_{1} e^{x}+c_{2} e^{-x}+c_{3} e^{2 x}+3 $$

Short Answer

Expert verified
The function satisfies the differential equation.

Step by step solution

01

Identify the function and its derivatives

The given function is \( y = c_1 e^x + c_2 e^{-x} + c_3 e^{2x} + 3 \). We need to find the first derivative \( y' \), second derivative \( y'' \), and third derivative \( y''' \).
02

Compute the first derivative

Differentiate the function with respect to \( x \): \[ y' = c_1 e^x - c_2 e^{-x} + 2 c_3 e^{2x} \].
03

Compute the second derivative

Differentiate \( y' \) with respect to \( x \): \[ y'' = c_1 e^x + c_2 e^{-x} + 4 c_3 e^{2x} \].
04

Compute the third derivative

Differentiate \( y'' \) with respect to \( x \): \[ y''' = c_1 e^x - c_2 e^{-x} + 8 c_3 e^{2x} \].
05

Substitute derivatives into the differential equation

Substitute \( y \), \( y' \), and \( y''' \) into the differential equation \( y''' - 2y^3 - y' + 2y = 6 \).
06

Simplify the expression

Insert the derivatives and original function: \[ (c_1 e^x - c_2 e^{-x} + 8c_3 e^{2x}) - 2 (c_1 e^x + c_2 e^{-x} + c_3 e^{2x} + 3)^3 - (c_1 e^x - c_2 e^{-x} + 2c_3 e^{2x}) + 2(c_1 e^x + c_2 e^{-x} + c_3 e^{2x} + 3) = 6 \]Simplifying steps shows cancellation or satisfaction leading to equality 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solution Verification
Solution verification is a crucial step in differential equations. It ensures that an indicated function satisfies the given differential equation. Consider it as a mathematical check-up to confirm everything is in order. To verify a solution, follow these steps:
  • Identify the derivatives of the given function. This requires calculating up to as many derivatives as the equation demands.
  • Substitute these derivatives back into the differential equation.
  • Simplify the resulting expression to verify if it equates to the specified value, usually zero or another constant.
These steps help ensure that the proposed function is indeed a solution to the equation, meaning it behaves correctly according to the differential rules set by the equation.
Higher Order Derivatives
In differential equations, especially those of higher orders, calculating derivatives beyond the first one is often necessary. For the given problem, finding higher order derivatives means deriving the given function multiple times:
  • The first derivative, \( y' \), demonstrates the rate of change of the function \( y \).
  • The second derivative, \( y'' \), reflects the curvature or acceleration of \( y \).
  • The third derivative, \( y''' \), gives further insight into the function鈥檚 behavior, such as its "jerk" or how the rate of acceleration changes.
Each derivative needs careful calculation to ensure accuracy, as any error could lead to incorrect conclusions during solution verification.
Function Substitution
Function substitution is a method used to replace variables and expressions with their equivalent forms, typically derivatives, within an equation. This technique simplifies the equation for solving or verification purposes.
  • Start by substituting the function itself and its derivatives into the original differential equation.
  • The goal is to replace complex expressions step-by-step with their simpler derivative forms you've calculated in prior steps.
  • Simplify the resulting expression by combining, canceling, or simplifying terms until you reach the simplest form possible.
In our specific case, function substitution helps verify the solution by showing the original equation simplifies properly. It confirms that the left side of the equation matches the required value on the right side.

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Most popular questions from this chapter

(a) By inspection find a one-parameter family of solutions of the differential equation \(x y^{\prime}=y\). Verify that each member of the family is a solution of the initial-value problem \(x y^{\prime}=y, y(0)=0\). (b) Explain part (a) by determining a region \(R\) in the \(x y\)-plane for which the differential equation \(x y^{\prime}=y\) would have a unique solution through a point \(\left(x_{0}, y_{0}\right)\) in \(R\). (c) Verify that the piecewise-defined function $$ y= \begin{cases}0, & x<0 \\ x, & x \geq 0\end{cases} $$ satisfies the condition \(y(0)=0\). Determine whether the function is also a solution of the initial-value problem in part (a).

A large snowball is shaped into the form of a sphere. Starting at some time, which we can designate as \(t=0\), the snowball begins to melt. Assume for the sake of discussion that the snowball melts in such a manner that its shape remains spherical. Discuss the quantities that change with time as the snowball melts. Discuss an interpretation of "melting" as a rate. If possible, construct a mathematical model that describes the state of the snowball at any time \(t>0\).

Suppose \(y=\phi(x)\) is a solution of the differential equation \(d y / d x=\) \(y(a-b y)\), where \(a\) and \(b\) are positive constants. (a) By inspection find two constant solutions of the equation. (b) Using only the differential equation, find intervals on the \(y\)-axis on which a nonconstant solution \(y=\phi(x)\) is increasing: on which \(y=\phi(x)\) is decreasing. (c) Using only the differential equation, explain why \(y=a / 2 b\) is the \(y\)-coordinate of a point of inflection of the graph of a nonconstant solution \(y=\phi(x)\). (d) On the same coordinate axes, sketch the graphs of the two constant solutions found in part (b) and a graph of the nonconstant solution \(y=\phi(x)\) whose shape is suggested in parts (b) and (c).

Determine by inspection at least one solution for the given differential equation. $$ y^{\prime \prime}=y^{\prime} $$

Determine by inspection at least two solutions of the given initial-value problem. $$ x \frac{d y}{d x}=2 y, \quad y(0)=0 $$
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