Introduction

Welcome to this comprehensive guide on verifying solutions for differential equations! This topic teaches you how to confirm whether a given function actually satisfies a differential equation, which is fundamental to ensuring that proposed solutions are mathematically correct.

The essential knowledge tells us that derivatives can be used to verify that a function is a solution to a given differential equation. Additionally, we learn that there may be infinitely many general solutions to a differential equation, which introduces us to the concept of families of solutions containing arbitrary constants.

Understanding Solutions to Differential Equations

A solution to a differential equation is a function that, when substituted into the equation along with its derivatives, makes the equation true for all values in its domain.

For a first-order differential equation $\frac{dy}{dx}=f(x,y)$, we say that $y=g(x)$ is a solution if:

$\frac{d}{dx}[g(x)]=f(x,g(x))$

Types of Solutions

General Solution: Contains one or more arbitrary constants and represents a family of functions that satisfy the differential equation.

Particular Solution: A specific function obtained by assigning particular values to the arbitrary constants, often determined by initial conditions.

Example 1:

Consider the differential equation $\frac{dy}{dx}=2x$.

The general solution is $y=x^2+C$, where $C$ is an arbitrary constant.

A particular solution might be $y=x^2+3$ (when $C=3$).

The Verification Process

To verify that a function is a solution to a differential equation, follow these systematic steps:

Step 1: Identify the Proposed Solution

Write down the function you're testing: $y=g(x)$.

Step 2: Compute Required Derivatives

Calculate the derivatives needed for the differential equation. For first-order equations, you need $\frac{dy}{dx}$.

Step 3: Substitute into the Differential Equation

Replace $y$ and its derivatives in the original equation with the expressions from your proposed solution.

Step 4: Simplify and Verify

Simplify both sides of the equation and check if they are equal.

Example 2:

Verify that $y=3e^{2x}$ is a solution to $\frac{dy}{dx}=2y$.

Step 1: Proposed solution: $y=3e^{2x}$

Step 2: Compute the derivative: $\frac{dy}{dx}=\frac{d}{dx}[3e^{2x}]=3\cdot 2e^{2x}=6e^{2x}$

Step 3: Substitute into $\frac{dy}{dx}=2y$:

Step 4: Since both sides equal $6e^{2x}$, the function is indeed a solution.

Verifying General Solutions

General solutions contain arbitrary constants and represent families of solutions.

Example 3:

Verify that $y=C{e}^{3x}$ is the general solution to $\frac{dy}{dx}=3y$.

Step 1: Proposed general solution: $y=C{e}^{3x}$

Step 2: Compute the derivative: $\frac{dy}{dx}=\frac{d}{dx}[C{e}^{3x}]=C\cdot 3e^{3x}=3C{e}^{3x}$

Step 3: Substitute into $\frac{dy}{dx}=3y$:

Step 4: Both sides are equal, confirming that $y=C{e}^{3x}$ is the general solution for any constant $C$.

Verifying Solutions with Initial Conditions

When initial conditions are given, we can verify both the general solution and find the specific constant value.

Example 4:

Verify that $y=C{e}^{-x}+2$ satisfies $\frac{dy}{dx}+y=2$ with initial condition $y(0)=5$.

Verifying the differential equation:

Step 1: Proposed solution: $y=C{e}^{-x}+2$

Step 2: Compute the derivative: $\frac{dy}{dx}=\frac{d}{dx}[C{e}^{-x}+2]=C(-1)e^{-x}+0=-C{e}^{-x}$

Step 3: Substitute into $\frac{dy}{dx}+y=2$:

Step 4: Both sides equal 2, so the solution satisfies the differential equation.

Verifying the initial condition: $y(0)=C{e}^{-0}+2=C+2=5$ Therefore, $C=3$, and the particular solution is $y=3e^{-x}+2$.

Verifying Higher-Order Equations

For second-order differential equations, we need to compute both first and second derivatives.

Example 5:

Verify that $y=A\cos (2x)+B\sin (2x)$ is the general solution to $\frac{d^{2}y}{d{x}^2}+4y=0$.

Step 1: Proposed solution: $y=A\cos (2x)+B\sin (2x)$

Step 2: Compute derivatives: $\frac{dy}{dx}=-2A\sin (2x)+2B\cos (2x)$

$\frac{d^{2}y}{d{x}^2}=-4A\cos (2x)-4B\sin (2x)=-4[A\cos (2x)+B\sin (2x)]=-4y$

Step 3: Substitute into $\frac{d^{2}y}{d{x}^2}+4y=0$: $-4y+4y=0$

Step 4: The equation is satisfied, confirming the general solution.

Common Verification Mistakes

Mistake 1: Incorrect Derivative Calculation

Always double-check derivative computations, especially with exponential and trigonometric functions.

Mistake 2: Algebraic Errors During Substitution

Carefully substitute expressions and simplify step by step.

Mistake 3: Forgetting Chain Rule

When dealing with composite functions, remember to apply the chain rule correctly.

Example 6:

For $y=e^{x^2}$, the derivative is $\frac{dy}{dx}=e^{x^2}\cdot 2x=2x{e}^{x^2}$, not just $e^{x^2}$.

Infinitely Many Solutions

Most differential equations have infinitely many solutions because of arbitrary constants in the general solution.

Example 7:

The differential equation $\frac{dy}{dx}=x$ has the general solution $y=\frac{x^2}{2}+C$.

Each value of $C$ gives a different particular solution:

All of these functions satisfy the original differential equation, illustrating the concept of a family of solutions.

Verification with Parametric Solutions

Some differential equations have solutions involving parameters other than simple constants.

Example 8:

Verify that $y=x+\frac{1}{x}$ is a solution to $x\frac{dy}{dx}+y=2x$.

Step 1: Proposed solution: $y=x+\frac{1}{x}$

Step 2: Compute the derivative: $\frac{dy}{dx}=1-\frac{1}{x^2}$

Step 3: Substitute into $x\frac{dy}{dx}+y=2x$:

Step 4: Both sides equal $2x$, confirming the solution.

Practice Section