How To Solve First Order Differential Equations

Solving first-order differential equations unlocks a powerful set of tools for modeling and understanding change in a multitude of real-world scenarios, from population growth and radioactive decay to the motion of objects and the flow of current in electrical circuits. Mastering the techniques to tackle these equations provides a fundamental building block for more advanced mathematical and scientific studies.

Understanding First-Order Differential Equations

A first-order differential equation is an equation that involves an unknown function and its first derivative. It can generally be expressed in the form:

dy/dx = f(x, y)

where:

• y is the unknown function of x (i.e., y = y(x)).
• dy/dx represents the first derivative of y with respect to x.
• f(x, y) is a function of both x and y.

Key Concepts:

• Order: The order of a differential equation is determined by the highest derivative present. Since we're dealing with dy/dx, this is a first-order equation.

• Linearity: A first-order differential equation is linear if it can be written in the form:

  dy/dx + P(x)y = Q(x)

  where P(x) and Q(x) are functions of x only. Otherwise, it's considered nonlinear. Linearity simplifies the solution process.

• Solution: A solution to a differential equation is a function y(x) that, when substituted into the equation, satisfies the equation for all values of x in a given interval.

• General Solution: The general solution contains an arbitrary constant (usually denoted as 'C'). This constant arises from the integration process and represents a family of solutions.

• Particular Solution: A particular solution is obtained from the general solution by specifying a particular value for the arbitrary constant 'C'. This is usually achieved by using an initial condition (e.g., y(x₀) = y₀).

Methods for Solving First-Order Differential Equations

Several methods exist for solving first-order differential equations. The most common and useful include:

1. Separation of Variables: This is the simplest and most direct method, applicable when the equation can be rearranged so that all terms involving y are on one side and all terms involving x are on the other.

2. Integrating Factor: This method is used for linear first-order differential equations. It involves multiplying the entire equation by a carefully chosen function (the integrating factor) to make the left-hand side a perfect derivative.

3. Exact Equations: This method applies when the differential equation can be written in a specific form where the left-hand side is the exact differential of a function of two variables.

4. Homogeneous Equations: These equations have a specific structure that allows them to be transformed into separable equations through a substitution.

5. Bernoulli Equations: These are nonlinear equations that can be transformed into linear equations using a suitable substitution.

Let's explore each of these methods in detail.

1. Separation of Variables

This method is applicable to equations of the form:

dy/dx = g(x)h(y)

Steps:

1. Separate the variables: Rewrite the equation so that all y terms are on one side with dy, and all x terms are on the other side with dx. This yields:

   dy/h(y) = g(x) dx

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

   ∫ [dy/h(y)] = ∫ g(x) dx

3. Solve for y: After integration, solve the resulting equation for y in terms of x. This will give you the general solution. Remember to include the constant of integration, C.

Example:

Solve the differential equation: dy/dx = x/y

1. Separate variables: y dy = x dx

2. Integrate both sides: ∫ y dy = ∫ x dx This gives: (y²/2) = (x²/2) + C

3. Solve for y: y² = x² + 2C. Let K = 2C, then y² = x² + K. Finally, y = ±√(x² + K)

This is the general solution. To find a particular solution, you would need an initial condition, such as y(0) = 2. Substituting these values into the general solution, we get 2 = ±√(0² + K), so 4 = K. Therefore, the particular solution is y = √(x² + 4) (we choose the positive square root since y(0) = 2 is positive).

2. Integrating Factor Method

This method is used for linear first-order differential equations of the form:

dy/dx + P(x)y = Q(x)

Steps:

1. Identify P(x) and Q(x): Make sure the equation is in the standard linear form and identify the functions P(x) and Q(x).

2. Calculate the integrating factor: The integrating factor, denoted by μ(x), is given by:

   μ(x) = e^(∫P(x) dx)

3. Multiply the equation by the integrating factor: Multiply both sides of the original differential equation by μ(x):

   μ(x) (dy/dx + P(x)y) = μ(x)Q(x)

4. Recognize the left-hand side as a derivative: The left-hand side should now be the derivative of the product of y(x) and the integrating factor:

   d/dx [μ(x)y(x)] = μ(x)Q(x)

5. Integrate both sides: Integrate both sides of the equation with respect to x:

   ∫ d/dx [μ(x)y(x)] dx = ∫ μ(x)Q(x) dx This simplifies to: μ(x)y(x) = ∫ μ(x)Q(x) dx + C

6. Solve for y: Divide both sides by μ(x) to obtain the solution for y:

   y(x) = (1/μ(x)) [∫ μ(x)Q(x) dx + C]

Example:

Solve the differential equation: dy/dx + 2y = e^(-x)

1. Identify P(x) and Q(x): P(x) = 2 and Q(x) = e^(-x)

2. Calculate the integrating factor: μ(x) = e^(∫2 dx) = e^(2x)

3. Multiply the equation by the integrating factor: e^(2x) (dy/dx + 2y) = e^(2x)e^(-x) which simplifies to e^(2x) (dy/dx + 2y) = e^(x)

4. Recognize the left-hand side as a derivative: d/dx [e^(2x)y(x)] = e^(x)

5. Integrate both sides: ∫ d/dx [e^(2x)y(x)] dx = ∫ e^(x) dx This gives: e^(2x)y(x) = e^(x) + C

6. Solve for y: y(x) = (e^(x) + C) / e^(2x) which simplifies to y(x) = e^(-x) + Ce^(-2x)

This is the general solution.

3. Exact Equations

A first-order differential equation of the form

M(x, y) dx + N(x, y) dy = 0

is said to be exact if there exists a function F(x, y) such that:

∂F/∂x = M(x, y) and ∂F/∂y = N(x, y)

Test for Exactness:

The necessary and sufficient condition for exactness is:

∂M/∂y = ∂N/∂x

Steps to Solve an Exact Equation:

1. Check for Exactness: Verify that ∂M/∂y = ∂N/∂x. If this condition is not satisfied, the equation is not exact, and this method cannot be used directly.

2. Find F(x, y): Integrate M(x, y) with respect to x, treating y as a constant:

   F(x, y) = ∫ M(x, y) dx + g(y)

   where g(y) is an arbitrary function of y only. This is because the partial derivative of a function of y with respect to x is zero.

3. Determine g(y): Differentiate the expression for F(x, y) obtained in step 2 with respect to y:

   ∂F/∂y = ∂/∂y [∫ M(x, y) dx + g(y)] = N(x, y)

   Solve this equation for g'(y) (the derivative of g(y) with respect to y) and then integrate to find g(y). Remember to include a constant of integration.

4. Write the general solution: The general solution to the exact equation is given by:

   F(x, y) = C

   where C is an arbitrary constant.

Example:

Solve the differential equation: (2xy + y²) dx + (x² + 2xy) dy = 0

1. Check for Exactness: M(x, y) = 2xy + y² and N(x, y) = x² + 2xy ∂M/∂y = 2x + 2y and ∂N/∂x = 2x + 2y Since ∂M/∂y = ∂N/∂x, the equation is exact.

2. Find F(x, y): F(x, y) = ∫ (2xy + y²) dx = x²y + xy² + g(y)

3. Determine g(y): ∂F/∂y = x² + 2xy + g'(y) = N(x, y) = x² + 2xy Therefore, g'(y) = 0, which implies g(y) = K (a constant). We can absorb this constant into the final constant C.

4. Write the general solution: x²y + xy² = C

4. Homogeneous Equations

A first-order differential equation is homogeneous if it can be written in the form:

dy/dx = f(y/x)

This means that the function f depends only on the ratio y/x.

Steps to Solve a Homogeneous Equation:

1. Verify Homogeneity: Check if the equation can be written in the form dy/dx = f(y/x).

2. Make the Substitution: Let v = y/x, which implies y = vx. Differentiate this with respect to x using the product rule:

   dy/dx = v + x(dv/dx)

3. Substitute into the Original Equation: Replace y/x with v and dy/dx with v + x(dv/dx) in the original differential equation:

   v + x(dv/dx) = f(v)

4. Separate Variables: Rearrange the equation and separate the variables v and x:

   x(dv/dx) = f(v) - v dv / (f(v) - v) = dx / x

5. Integrate Both Sides: Integrate both sides of the separated equation:

   ∫ dv / (f(v) - v) = ∫ dx / x

6. Solve for v: Solve the resulting equation for v in terms of x.

7. Substitute Back: Replace v with y/x to obtain the solution in terms of x and y.

Example:

Solve the differential equation: dy/dx = (x² + y²) / (xy)

1. Verify Homogeneity: Rewrite the equation as: dy/dx = (x/y) + (y/x). This is a function of y/x, so it's homogeneous. We can also write it as dy/dx = x/y + y/x

2. Make the Substitution: Let v = y/x, so y = vx and dy/dx = v + x(dv/dx)

3. Substitute into the Original Equation: v + x(dv/dx) = x/(vx) + vx/x which simplifies to v + x(dv/dx) = 1/v + v

4. Separate Variables: x(dv/dx) = 1/v. Therefore, v dv = dx/x

5. Integrate Both Sides: ∫ v dv = ∫ dx/x. This gives v²/2 = ln|x| + C

6. Solve for v: v² = 2ln|x| + 2C

7. Substitute Back: (y/x)² = 2ln|x| + K (where K = 2C). Therefore, y² = x²(2ln|x| + K) and y = ±x√(2ln|x| + K)

5. Bernoulli Equations

A Bernoulli equation is a nonlinear differential equation of the form:

dy/dx + P(x)y = Q(x)y^n

where n is a real number (but not 0 or 1, as those cases would make the equation linear).

Steps to Solve a Bernoulli Equation:

1. Divide by y^n: Divide the entire equation by y^n:

   y^(-n) (dy/dx) + P(x)y^(1-n) = Q(x)

2. Make the Substitution: Let z = y^(1-n). Then, dz/dx = (1-n)y^(-n)(dy/dx). Solve for y^(-n)(dy/dx):

   y^(-n) (dy/dx) = (1/(1-n)) (dz/dx)

3. Substitute into the Modified Equation: Substitute z and (1/(1-n))(dz/dx) into the equation from step 1:

   (1/(1-n)) (dz/dx) + P(x)z = Q(x)

4. Simplify and Solve the Linear Equation: Multiply the equation by (1-n) to obtain a linear first-order differential equation in z:

   dz/dx + (1-n)P(x)z = (1-n)Q(x)

   Solve this linear equation using the integrating factor method.

5. Substitute Back: Replace z with y^(1-n) to obtain the solution in terms of x and y.

Example:

Solve the differential equation: dy/dx + y = xy³

1. Divide by y³: y^(-3) (dy/dx) + y^(-2) = x

2. Make the Substitution: Let z = y^(1-3) = y^(-2). Then, dz/dx = -2y^(-3) (dy/dx). Solve for y^(-3)(dy/dx): y^(-3)(dy/dx) = -1/2 (dz/dx)

3. Substitute into the Modified Equation: -1/2 (dz/dx) + z = x

4. Simplify and Solve the Linear Equation: dz/dx - 2z = -2x.

   • P(x) = -2, Q(x) = -2x
   • Integrating Factor: μ(x) = e^(∫-2 dx) = e^(-2x)
   • Multiply the equation by the integrating factor: e^(-2x)(dz/dx - 2z) = -2xe^(-2x)
   • Recognize the left-hand side as a derivative: d/dx [e^(-2x)z] = -2xe^(-2x)
   • Integrate both sides: ∫ d/dx [e^(-2x)z] dx = ∫ -2xe^(-2x) dx. Use integration by parts for the right-hand side: ∫ -2xe^(-2x) dx = xe^(-2x) + (1/2)e^(-2x) + C
   • Therefore, e^(-2x)z = xe^(-2x) + (1/2)e^(-2x) + C
   • Solve for z: z = x + 1/2 + Ce^(2x)

5. Substitute Back: y^(-2) = x + 1/2 + Ce^(2x). Therefore, y² = 1 / (x + 1/2 + Ce^(2x)) or y = ±√(1 / (x + 1/2 + Ce^(2x)))

Practical Applications

First-order differential equations are fundamental in modeling numerous real-world phenomena:

• Population Growth/Decay: The rate of change of a population is often proportional to the current population size, leading to an equation of the form dy/dt = ky, where y(t) is the population at time t, and k is a constant.

• Radioactive Decay: The decay rate of a radioactive substance is proportional to the amount of the substance present.

• Newton's Law of Cooling: The rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature.

• Mixing Problems: These problems involve determining the amount of a substance in a tank at a given time, where fluid is entering and leaving the tank.

• Electrical Circuits (RC and RL Circuits): The flow of current in simple circuits containing resistors and capacitors (RC circuits) or resistors and inductors (RL circuits) can be modeled using first-order differential equations.

• Chemical Reactions: The rate of a first-order chemical reaction is proportional to the concentration of the reactant.

Tips and Common Mistakes

• Don't forget the constant of integration! Always include "+ C" after performing an indefinite integral. This constant is crucial for representing the general solution.

• Check your work. After finding a solution, substitute it back into the original differential equation to verify that it satisfies the equation.

• Be careful with algebra. A common source of errors is algebraic manipulation. Double-check your steps to avoid mistakes.

• Understand the assumptions. Be aware of the assumptions underlying the models you are using. For example, the population growth model assumes that the growth rate is constant, which may not be realistic in the long term.

• Practice, practice, practice! The best way to master solving differential equations is to practice solving a variety of problems.

Conclusion

Solving first-order differential equations is a fundamental skill in mathematics, science, and engineering. By understanding the different types of equations and the appropriate solution methods, you can tackle a wide range of problems involving rates of change. Mastering techniques like separation of variables, integrating factors, and recognizing exact and homogeneous equations will significantly expand your problem-solving capabilities and provide a solid foundation for more advanced studies in differential equations and related fields. Remember to practice regularly and pay attention to detail to avoid common errors.