Finding General Solutions Using Separation of Variables - AP Calculus BC Study Guide

The Core Idea: Finding General Solutions Using Separation of Variables

The central task of this topic is to find the family of functions, known as the general solution, that satisfies a given differential equation. Not all differential equations can be solved with the methods learned in this course, but a specific class of them, called separable differential equations, can be. The method of separation of variables is an algebraic and calculus-based technique for solving these specific types of differential equations.

The entire process hinges on a key structural feature: the differential equation must be expressible in a form where the derivative, $dy/dx$, is equal to a product of two separate functions, one exclusively in terms of $x$ and the other exclusively in terms of $y$. If a differential equation can be written in this form, $dy/dx=g(x)h(y)$, we can algebraically "separate" the variables $x$ and $y$ to opposite sides of the equation. Once separated, we can integrate each side with respect to its own variable. This integration process introduces a constant of integration, $C$, which is why the result is a "general solution"—it represents an entire family of solution curves, each corresponding to a different value of $C$.

Key Procedure: Separation of Variables

The method of separation of variables is a procedural algorithm for solving differential equations of a specific form. The procedure is derived directly from the structure of a separable differential equation.

Given a differential equation of the form:

$$\frac{dy}{dx}=g(x)h(y)$$

The goal is to isolate all terms involving $y$ on one side of the equation with $dy$, and all terms involving $x$ on the other side with $dx$.

Step 1: Separate the Variables

Treat $dy/dx$ as a ratio and multiply and divide to move all $y$ terms to the left and all $x$ terms to the right.

$$\frac{1}{h(y)}dy=g(x)dx$$

This step is only valid if $h(y)\ne 0$.

Step 2: Integrate Both Sides

Integrate both sides of the separated equation with respect to their respective variables.

$$\int \frac{1}{h(y)}dy=\int g(x)dx$$

Step 3: Find the Antiderivatives and Add the Constant of Integration

Evaluate the integrals. Let $H(y)$ be the antiderivative of $1/h(y)$ and $G(x)$ be the antiderivative of $g(x)$.

$$H(y)=G(x)+C$$

Note that a constant of integration arises from both indefinite integrals. However, we can combine them into a single constant, $C$, typically placed on the side with the independent variable ($x$). This single constant accounts for all possible constants and defines the family of solutions. The resulting equation is the general solution in implicit form. If possible, this equation can be solved for $y$ to find the explicit general solution.

Understanding Separable Equations

The critical prerequisite for using this method is recognizing whether a differential equation is separable. The Essential Knowledge defines this form as $dy/dx=g(x)h(y)$. This means the expression for the derivative must be a product of a function of $x$ and a function of $y$.

Recognizing the Separable Form $g(x)h(y)$:

• Separable:$dy/dx=x^{2}y$

  • Here, $g(x)=x^2$ and $h(y)=y$. This fits the form.

• Separable:$dy/dx=\frac{\sin (x)}{y^2}$

  • This can be written as $dy/dx=\sin (x)\cdot \frac{1}{y^2}$. Here, $g(x)=\sin (x)$ and $h(y)=1/y^2$. This fits the form.

• Separable:$dy/dx=e^{x+y}$

  • Using properties of exponents, this can be rewritten as $dy/dx=e^x{e}^y$. Here, $g(x)=e^x$ and $h(y)=e^y$. This fits the form.

• Separable:$dy/dx=3y$

  • This can be written as $dy/dx=3\cdot y$. Here, $g(x)=3$ (a constant function of $x$) and $h(y)=y$. This fits the form.

Recognizing Non-Separable Forms:

• Not Separable:$dy/dx=x+y$

  • There is no way to factor this expression into a product of a function of $x$ only and a function of $y$ only. The addition prevents separation.

• Not Separable:$dy/dx=\sin (xy)$

  • The variables $x$ and $y$ are intertwined as the argument of the sine function and cannot be factored into a $g(x)h(y)$ form.

The ability to correctly identify the structure of the differential equation is the foundational skill for this topic. If an equation is not separable, the method of separation of variables cannot be applied.

Core Concepts & Rules

• Purpose of the Method: The method of separation of variables is an analytical technique used to find general solutions for a specific class of differential equations.

• Required Form: This method applies only to differential equations that can be written in the form $dy/dx=g(x)h(y)$, where the derivative is a product of a function of $x$ and a function of $y$.

• The Separation Step: The core of the method involves algebraically manipulating the equation to isolate all $y$ terms with $dy$ on one side and all $x$ terms with $dx$ on the other, yielding $(1/h(y))dy=g(x)dx$.

• The Integration Step: After separating, find the general solution by integrating both sides: $\int (1/h(y))dy=\text{intg}(x)dx$.

• General Solution and the Constant $C$: The process of indefinite integration introduces an arbitrary constant, $C$. The final equation, which includes this constant, is called the general solution because it represents the entire family of functions that satisfy the original differential equation.

Step-by-Step Example 1: Basic Application

Problem: Find the general solution to the differential equation $dy/dx=3x^{2}y$.

Step 1: Identify $g(x)$ and $h(y)$ and Separate the Variables

The equation is in the form $dy/dx=g(x)h(y)$ where $g(x)=3x^2$ and $h(y)=y$. We can now separate the variables.

$$\frac{dy}{dx}=3x^{2}y$$

Divide both sides by $y$ and multiply both sides by $dx$. Assume $y\ne 0$.

$$\frac{1}{y}dy=3x^{2}dx$$

Step 2: Integrate Both Sides

Integrate both sides of the separated equation.

$$\int \frac{1}{y}dy=\int 3x^{2}dx$$

Step 3: Evaluate the Integrals and Add the Constant of Integration

The integral of $1/y$ with respect to $y$ is $ln|y|$. The integral of $3x^2$ with respect to $x$ is $x^3$. We add a single constant of integration, $C$, to the side with the $x$ variable.

$$\ln |y|=x^3+C$$

This is the general solution in implicit form.

Step 4: (Optional) Solve for $y$ to find the Explicit Solution

To find the explicit solution, we can exponentiate both sides using base $e$.

$$e^{\ln |y|}=e^{x^3+C}$$

$$|y|=e^{x^3}\cdot e^C$$

Since $C$ is an arbitrary constant, $e^C$ is also a constant. Let's call this new constant $A=e^C$. Because $e^C$ must be positive, $A$ must be positive.

$$|y|=A{e}^{x^3}(A>0)$$

Removing the absolute value introduces a $\pm$ sign.

$$y=\pm A{e}^{x^3}$$

We can combine the $\pm$ sign with the positive constant $A$ to create a new arbitrary constant, let's call it $C_1$, which can be any non-zero real number ($C_1=\pm A$).

$$y=C_1{e}^{x^3}(C_1\ne 0)$$

We should also check if $y=0$ is a solution. Plugging $y=0$ into the original $dy/dx=3x^{2}y$ gives $0=3x^2(0)$, which is 0=0`. So $y=0 is also a solution. This corresponds to the case where $C_1=0$. Therefore, the final general solution is:

$$y=C{e}^{x^3}$$

where $C$ is any real number.

Step-by-Step Example 2: Exam-Style Application

Problem: Find the general solution to the differential equation $y^{\prime }=\frac{x\sin (x^2)}{y}$.

Step 1: Rewrite and Separate the Variables

First, write $y^{\prime }$ as $dy/dx$.

$$\frac{dy}{dx}=\frac{x\sin (x^2)}{y}$$

This equation is in the form $dy/dx=g(x)h(y)$ where $g(x)=x\sin (x^2)$ and $h(y)=1/y$.

Separate the variables by multiplying both sides by $y$ and $dx$. Assume $y\ne 0$.

$$ydy=x\sin (x^2)dx$$

Step 2: Integrate Both Sides

Set up the integrals for both sides.

$$\int ydy=\int x\sin (x^2)dx$$

Step 3: Evaluate the Integrals

The left side is a straightforward power rule integration.

$$\int ydy=\frac{1}{2}{y}^2$$

The right side requires a u-substitution. Let $u=x^2$. Then $du=2xdx$, which means $xdx=\frac{1}{2}du$.

$$\int x\sin (x^2)dx=\int \sin (u)\cdot \frac{1}{2}du$$

$$=\frac{1}{2}\int \sin (u)du=\frac{1}{2}(-\cos (u))=-\frac{1}{2}\cos (u)$$

Substitute back $u=x^2$:

$$\int x\sin (x^2)dx=-\frac{1}{2}\cos (x^2)$$

Step 4: Combine and Add the Constant of Integration

Now, set the results of the integrations equal to each other and add the constant $C$.

$$\frac{1}{2}{y}^2=-\frac{1}{2}\cos (x^2)+C$$

This is the general solution in implicit form. We can make it look cleaner by multiplying the entire equation by 2.

$$y^2=-\cos (x^2)+2C$$

Since $C$ is an arbitrary constant, 2C` is also just an arbitrary constant. We can rename it $K for simplicity.

$$y^2=-\cos (x^2)+K$$

Step 5: (Optional) Solve for $y$

To get the explicit solution, take the square root of both sides.

$$y=\pm \sqrt{K-\cos (x^2)}$$

This is the explicit general solution.

Using Your Calculator

The process of finding a general solution using separation of variables is a purely analytical technique. It relies on algebraic manipulation (separation) and symbolic integration. A graphing calculator cannot perform these symbolic steps for you to derive the general solution.

Therefore, this topic is typically tested on the no-calculator portion of the AP exam.

While a calculator cannot find the general solution $y=G(x)+C$, it can be used in a limited capacity to check your work for a particular solution (once a value for $C$ is known). For example, if you found a particular solution $y=f(x)$, you could:

1. Numerically calculate the derivative of your solution $f(x)$ at a point $x=a$ using the calculator's numerical derivative feature (e.g., nDeriv or $d/dx$).

2. Substitute $x=a$ and $y=f(a)$ into the original differential equation $dy/dx=g(x)h(y)$ and calculate the value.

3. If your solution is correct, the values from Step 1 and Step 2 should be equal.

This is a verification method, not a solution method. You must be able to perform the separation and integration by hand.

AP Exam Quick Hit

Common Question Types

• Find the General Solution: You will be given a separable differential equation and asked to find its general solution. This is the most direct test of the topic.

  • Example: "Find the general solution to the differential equation $dy/dx=e^y\cos (x)$."

• Identify a Separable Equation: In a multiple-choice question, you may be presented with several differential equations and asked to identify which one is separable. This tests your understanding of the required $dy/dx=g(x)h(y)$ structure.

  • Example: "Which of the following differential equations is separable?

    (A) $y^{\prime }=x^2+y^2$

    (B) $y^{\prime }=\frac{x+1}{y}$

    (C) $y^{\prime }=\ln (xy)$

    (D) $y^{\prime }=x-\sin (y)$"

    (Correct Answer: B, as it can be written $y^{\prime }=(x+1)\ast (1/y)$)

Common Mistakes

• Forgetting the Constant of Integration + C: The most common error is to perform the integration correctly but omit the + C. This finds only one particular solution, not the general solution.

• Incorrect Algebraic Separation: Errors in algebra when isolating variables. For example, incorrectly separating $dy/dx=x+y$ into $dy-y=xdx$. Remember, the form must be a product $g(x)h(y)$.

• Mishandling the Constant After Exponentiating: When solving for $y$ from an expression like $ln|y|=G(x)+C$, a common mistake is to write $y=e^{G(x)}+e^C$. The correct step is $|y|=e^{G(x)+C}=e^{G(x)}{e}^C$, where $e^C$ becomes a new multiplicative constant $A$.

• Placing + C Incorrectly: Adding a + C inside a function, such as writing $\ln |y+C|$ instead of $\ln |y|=...+C$. The constant is added after the antiderivative is found.

• Basic Integration Errors: Simple mistakes in finding the antiderivatives of $g(x)$ or $1/h(y)$, such as incorrect application of the power rule, u-substitution, or rules for trigonometric/logarithmic functions.