7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

A particular solution is the single function that satisfies a differential equation and passes through a given initial condition. You find it by separating variables, integrating both sides with a constant of integration, then using the initial point to solve for that constant. For AP Calculus, keep the constant until the initial condition determines its value.

Why This Matters for the AP Calculus Exam

Differential equations are a steady source of points on the AP Calculus AB and BC exams, and separation of variables with an initial condition is the core procedure. Free-response questions often give you a rate equation like $\frac{dy}{dt} = \frac{1}{5}(100 - B)$ plus a starting value, then ask for the particular solution. These problems reward a clean, predictable process: separate, integrate, add $C$, solve for $C$, and finish the algebra.

This topic also shows up in multiple-choice questions that ask you to identify a particular solution or evaluate it at a specific point. Showing each step clearly is important for earning the procedure-based points on free response, since partial credit often hinges on separating variables and including the constant.

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Key Takeaways

• A general solution is a family of functions with a constant $C$; a particular solution is the one curve that passes through a given point.
• Use the initial condition to solve for $C$ algebraically after integrating.
• The form $F(x) = y_0 + \int_a^x f(t)\, dt$ is a particular solution to $\frac{dy}{dx} = f(x)$ that satisfies $F(a) = y_0$.
• Separating variables and remembering $+C$ are the steps that earn early credit even if the integration gets tricky.
• Check for domain restrictions, such as division by zero, that limit where your solution is valid.
• Not every differential equation with a fraction gives a logarithm, so check what the integrand actually requires.

General Versus Particular Solutions

A general solution to a differential equation is a family of functions that all satisfy the equation. Because it contains a constant of integration, changing that constant produces infinitely many curves. You get a general solution when no initial condition is given, and plotting it gives a set of curves that follow the pattern of a slope field. For a refresher on slope fields, see 7.3 Sketching Slope Fields and 7.4 Reasoning Using Slope Fields.

A particular solution is the single curve that passes through a specific point. When the problem gives you an initial condition, you can pin down that one curve. There is only one particular solution through a given point.

For separation of variables review, check 7.6 Finding General Solutions Using Separation of Variables.

The Definite-Integral Form

When a free-response question asks you to write an expression for the solution and gives an initial condition, you can use this form:

This is a particular solution to $\frac{dy}{dx}=f(x)$ where $F(a)=y_0$ is the initial condition. It is useful when the integrand does not have a simple antiderivative, because you can leave the answer as an accumulation built from the starting value.

How to Use This on the AP Calculus Exam

Problem Solving

Follow the same routine every time:

1. Separate the variables. Gather all the $x$'s on one side and all the $y$'s on the other.
2. Integrate both sides with respect to their variables. Add $+C$.
3. Solve for $y$ to get the general solution.
4. Plug in the initial condition for $x$ and $y$, then solve for $C$.
5. Substitute $C$ back in and finish solving for the particular solution.

Watch the Domain

Solutions to differential equations are not always valid for every input. Restrictions can come from:

1. Singularities - values that cause division by zero or other undefined behavior.

2. Physical constraints - the solution may only make sense over a certain range, like a position that must stay above ground.

3. Mathematical constraints - the solution may only be valid for certain values, such as positive inputs.

Sometimes you need to restrict to one interval that contains the initial point. For example, suppose you want the particular solution of

with $y(3)=2$. The denominator is zero at $x=2$ and $x=-2$, so the solution cannot include those values. Since the initial condition starts at $x=3$, the valid interval is $x>2$.

Worked Free Response Example

This is part of a free-response question from the 2012 AP Calculus AB exam administered by College Board. We will work part (c), which focuses on solving with an initial condition.

The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time $t=0$, the bird weighs 20 grams. If $B(t)$ is the weight in grams at time $t$ days, then

Let $y=B(t)$ be the solution with initial condition $B(0)=20$.

(c) Use separation of variables to find $y=B(t)$, the particular solution with initial condition $B(0)=20$.

Step 1: Separate the $B$'s and $t$'s.

Step 2: Integrate both sides.

Step 3: Add $+C$ and write the general solution.

The single $C$ absorbs the constants from both integrals.

Step 4: Use $B(0)=20$ to solve for $C$.

The absolute value drops because the weight values stay positive over the relevant interval, so $100-B>0$.

Step 5: Substitute $C$ and solve for $B$.

Everything through Step 4 is calculus; the rest is algebra and arithmetic.

Common Trap

On a free-response differential equation, most of the points come from the differential equation work itself.

• Always separate your variables. Separating can earn a point even if you cannot finish the integration. Failing to separate usually loses the whole part.
• Never forget the $+C$. The constant is how you turn a general solution into a particular one using the initial condition. Without it, the most you can usually earn is two points: one for separating and one for the correct antiderivatives.

Common Misconceptions

• General and particular solutions are the same thing. A general solution has a constant and represents infinitely many curves. A particular solution uses the initial condition to lock in one specific curve.
• You can drop the constant of integration. Without $+C$, you cannot apply the initial condition, and you lose most of the available credit.
• Every fraction integrates to a logarithm. Only integrands of the form $\frac{1}{u}\,du$ give a natural log. Others may need a substitution, power rule, or different technique, so check before assuming.
• The absolute value in $\ln|y|$ can always be dropped. Only remove it when the context tells you the inside stays positive (or negative). Otherwise it can change your solution.
• A solution is valid everywhere. Domain restrictions from division by zero, physical limits, or required input conditions can shrink the interval where your particular solution applies. Keep the interval that contains the initial point.
• You should solve for $C$ before integrating. You can only find $C$ after you integrate and have the general solution to plug the initial condition into.

Related AP Calculus Guides

• Unit 7 Overview: Differential Equations
• 7.2 Verifying Solutions for Differential Equations
• 7.3 Sketching Slope Fields
• 7.6 Finding General Solutions Using Separation of Variables
• 7.5 Approximating Solutions Using Euler’s Method
• 7.4 Reasoning Using Slope Fields