Differential Equations

The One Thing

First-order differential equations are formed by differentiating a family of curves and eliminating its arbitrary constants, then solved by identifying the appropriate structure. The principal methods are separation of variables, the substitution $y=vx$ for homogeneous equations, and integrating factors for linear equations.

Definitions and Results

• Differential equation: An equation involving a dependent variable, an independent variable, and one or more derivatives of the dependent variable.
• Order: The highest order derivative present in a differential equation.
• Degree: The power of the highest order derivative after the equation has been made polynomial in derivatives.
• First-order differential equation: A differential equation involving only the first derivative of the dependent variable, such as

$$\frac{dy}{dx}=f(x,y).$$ Its general form is $$F\left(x,y,\frac{dy}{dx}\right)=0.$$

• General solution: A solution containing an arbitrary constant and therefore representing a family of curves.
• Particular solution: A solution obtained from the general solution by applying an initial or boundary condition.
• Formation of a differential equation: The process of differentiating a given family of equations and eliminating its arbitrary constants. If a family contains one arbitrary constant, differentiating once is generally sufficient. If it contains $n$ independent arbitrary constants, it generally requires $n$ differentiations to form a differential equation of order $n$.
• Initial condition: A specified value of the dependent variable for a particular value of the independent variable, used to determine the arbitrary constant.
• Variable-separable equation: An equation that can be rearranged into

$$g(y)dy=f(x)dx.$$

• Homogeneous first-order equation: An equation of the form

$$\frac{dy}{dx}=F\left(\frac{y}{x}\right),$$ usually solved using $y=vx$, where $v$ is a function of $x$. If the equation involves $x/y$, the substitution $x=vy$ may be used.

• Linear differential equation: A first-order equation in the standard form

$$\frac{dy}{dx}+P(x)y=Q(x).$$

• Integrating factor: For

$$\frac{dy}{dx}+P(x)y=Q(x),$$ the integrating factor is $$\mathrm{IF}=e^{\int P(x)dx}.$$ It converts the left-hand side into the derivative of a product.

• Exact differential equation: An equation written as

$$M(x,y)dx+N(x,y)dy=0$$ is exact when $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.$$

• Implicit and explicit solutions: A solution may be implicit, such as

$$x^2+y^2=C,$$ or explicit, such as $$y=f(x).$$

• Singular solution: A solution that may not be represented by the general solution and should therefore be checked separately when appropriate.

Worked Methods

1. Forming a differential equation by eliminating an arbitrary constant

• Start with a family containing an arbitrary constant, for example

$$y=f(x,C).$$

• Differentiate with respect to $x$.
• Use the original equation and its derivative to eliminate $C$.
• The resulting equation is the required differential equation.

Differentiation reduces the number of arbitrary constants in a family of curves. Thus, one arbitrary constant generally requires one differentiation, while $n$ independent arbitrary constants generally require $n$ differentiations to form an equation of order $n$.

2. Solving a variable-separable equation

For an equation of the form $$\frac{dy}{dx}=f(x)g(y):$$

• Rearrange the equation so that all terms involving $y$ are on one side:

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

• Integrate both sides:

$$\int \frac{dy}{g(y)}=\int f(x)dx.$$

• Include the constant of integration:

$$\text{integrated expression}=C.$$

• Use an initial condition, if provided, to determine $C$.
• Verify the result by differentiating it and substituting it into the original differential equation.

The solution may remain implicit, as in $$x^2+y^2=C,$$ unless it can be rearranged into an explicit form $y=f(x)$.

3. Solving a homogeneous first-order equation

For $$\frac{dy}{dx}=F\left(\frac{y}{x}\right):$$

• Substitute

$$y=vx.$$

• Differentiate:

$$\frac{dy}{dx}=v+x\frac{dv}{dx}.$$

• Substitute $y=vx$ and

$$\frac{dy}{dx}=v+x\frac{dv}{dx}$$ into the original equation.

• Rearrange the resulting equation, usually into a separable equation in $v$ and $x$.
• Integrate both sides and include the constant of integration.
• Replace $v$ by $y/x$.
• Apply any initial condition and verify the result.

When the equation is naturally expressed in terms of $x/y$, the alternative substitution $$x=vy$$ may be used.

4. Solving a linear differential equation using an integrating factor

For the standard linear equation $$\frac{dy}{dx}+P(x)y=Q(x):$$

• Identify $P(x)$ and $Q(x)$.
• Calculate the integrating factor:

$$\mathrm{IF}=e^{\int P(x)dx}.$$

• Multiply every term of the differential equation by the integrating factor.
• Recognise the left-hand side as a product derivative:

$$\frac{d}{dx}[y\times \mathrm{IF}]=Q(x)\times \mathrm{IF}.$$

• Integrate:

$$y\times \mathrm{IF}=\int Q(x)\times \mathrm{IF}dx+C.$$

• Solve for $y$, if an explicit solution is required.
• Use any initial condition to determine $C$.
• Verify the result by differentiation and substitution into the original equation.

5. Solving an exact differential equation

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

• Identify $M(x,y)$ and $N(x,y)$.
• Test exactness by evaluating

$$\frac{\partial M}{\partial y}\text{and}\frac{\partial N}{\partial x}.$$

• Confirm that

$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.$$

• Integrate $M$ with respect to $x$, or $N$ with respect to $y$.
• Combine the resulting terms to obtain the implicit solution.
• Include the constant of integration unless it has already been absorbed into another constant.
• Verify the solution by differentiating and substituting back into the original equation.

6. Obtaining a particular solution

• Find the general solution, including its arbitrary constant.
• Substitute the given initial or boundary condition.
• Solve for the arbitrary constant.
• Substitute the constant back into the general solution.
• The resulting equation is the particular solution.

The general solution describes all members of a family, whereas the initial condition selects one particular member.

Where It Goes Wrong

• Failing to eliminate the arbitrary constant after differentiating when forming a differential equation.
• Misidentifying the equation type instead of checking whether variables can be separated, whether the equation has the form $F(y/x)$, or whether it is linear.
• For a homogeneous equation, omitting

$$\frac{dy}{dx}=v+x\frac{dv}{dx}$$ after using $y=vx$.

• Using the wrong integrating factor instead of

$$e^{\int P(x)dx}$$ for the standard linear form.

• Omitting the constant of integration, unless it has genuinely been absorbed into another constant.
• Failing to apply the initial condition or to verify the obtained solution by differentiation and substitution; singular solutions may also need to be checked separately.

What Gets Asked

This material supports questions requiring students to:

• Define the differential equation, order, degree, first-order equation, general solution, particular solution, initial condition, homogeneous equation, linear equation, integrating factor, and exact equation.
• Form a differential equation from a family $y=f(x,C)$ by differentiation and elimination of arbitrary constants.
• Determine the order and degree of a differential equation.
• Solve separable equations by rearranging them as

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

• Solve homogeneous equations using $y=vx$, or $x=vy$ when the equation involves $x/y$.
• Solve linear equations using

$$\mathrm{IF}=e^{\int P(x)dx}.$$

• Test and solve exact equations satisfying

$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.$$

• Use an initial condition to obtain a particular solution from a general solution.
• Distinguish between implicit and explicit solutions.
• Verify a proposed solution by differentiation and substitution, including consideration of possible singular solutions.
• Explain how differential equations model growth, decay, motion, population, and other rates of change.