Differential Equations

The One Thing

Differential equations are formed by eliminating arbitrary constants from a family of relations and are solved by finding a relation between the variables that satisfies the equation. For first-order, first-degree equations, 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 derivatives of the dependent variable.

• Order: The order of the highest derivative occurring in a differential equation.

• Degree: The power of the highest-order derivative when the equation is polynomial in its derivatives, after removing radicals and fractions involving derivatives.

• First-order, first-degree differential equation: A differential equation containing only the first derivative, with that derivative occurring to the first power.

• General solution: A solution containing as many arbitrary constants as the order of the differential equation.

• Particular solution: A solution obtained from the general solution by assigning specific values to arbitrary constants using given initial or boundary conditions.

• Formation of a differential equation: The process of differentiating a family of curves and eliminating its arbitrary constants. If a relation contains $n$ arbitrary constants, it is differentiated $n$ times and the constants are eliminated, producing a differential equation of order $n$.

• Variable separable equation: A differential equation that can be rearranged so that all terms involving one variable occur on one side and all terms involving the other variable occur on the other side. In particular,

$$\frac{dy}{dx}=f(x)g(y)$$ can be written as $$\frac{dy}{g(y)}=f(x)dx.$$

• Homogeneous differential equation: A first-order equation of the form

$$\frac{dy}{dx}=F\left(\frac{y}{x}\right),$$ or an equation in which the numerator and denominator are homogeneous functions of the same degree.

• Substitution for homogeneous equations: Use

$$y=vx,$$ where $v$ is a function of $x$. Then $$\frac{dy}{dx}=v+x\frac{dv}{dx}.$$ The substitution $x=vy$ may also be used where appropriate.

• Linear differential equation: A first-order equation expressible as

$$\frac{dy}{dx}+P(x)y=Q(x),$$ where $P(x)$ and $Q(x)$ are functions of $x$. An equation in $x$ and $y$ may alternatively be written as $$\frac{dx}{dy}+P(y)x=Q(y),$$ with $x$ treated as the dependent variable.

• Integrating factor: For

$$\frac{dy}{dx}+P(x)y=Q(x),$$ the integrating factor is $$I.F.=e^{\int P(x)dx}.$$ The solution is $$y\times I.F.=\int [Q(x)\times I.F.]dx+C.$$

• Exact differential equation: An equation

$$M(x,y)dx+N(x,y)dy=0$$ is exact if $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.$$ Its solution is obtained by finding a function whose total differential equals the given expression.

• Solution of $\frac{dy}{dx}=f(x)$:

$$y=\int f(x)dx+C.$$ The constant of integration must be included.

Worked Methods

1. Formation by eliminating arbitrary constants

• Begin with a relation involving arbitrary constants.
• Differentiate sufficiently many times to obtain as many equations as there are arbitrary constants.
• Eliminate the arbitrary constants.
• Write the resulting differential equation.

For $$y=a{e}^x+b{e}^{-x},$$ where $a$ and $b$ are arbitrary constants:

$$\frac{dy}{dx}=a{e}^x-b{e}^{-x},$$

and differentiating again gives

$$\frac{d^{2}y}{d{x}^2}=a{e}^x+b{e}^{-x}.$$

Since $$y=a{e}^x+b{e}^{-x},$$ eliminating $a$ and $b$ gives

$$\boxed{\frac{d^{2}y}{d{x}^2}-y=0}.$$

The resulting equation is of order $2$, corresponding to the two arbitrary constants.

2. Separation of variables

• Identify whether the equation can be written in the form

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

• Rearrange it as

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

• Integrate both sides.
• Include the constant of integration.
• If a condition is supplied, substitute it into the general solution to determine the constant.

The general separated form is therefore

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

For the special case

$$\frac{dy}{dx}=f(x),$$

the solution is

$$\boxed{y=\int f(x)dx+C}.$$

3. Homogeneous equations

• Confirm that the equation has the form

$$\frac{dy}{dx}=F\left(\frac{y}{x}\right),$$ or that its numerator and denominator are homogeneous functions of the same degree.

• Substitute

$$y=vx.$$

• Differentiate the substitution:

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

• Substitute these expressions into the original equation.
• Rearrange the resulting equation so that the variables $v$ and $x$ are separated.
• Integrate.
• Replace $v$ by

$$\frac{y}{x}.$$

The essential transformation is

$$y=vx,\frac{dy}{dx}=v+x\frac{dv}{dx},$$

which changes the homogeneous equation into a variable separable equation.

4. Linear equations using an integrating factor

• Express the equation in the standard form

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

• Calculate the integrating factor:

$$I.F.=e^{\int P(x)dx}.$$

• Multiply the entire differential equation by the integrating factor.
• Integrate the resulting equation.
• Use

$$y\times I.F.=\int [Q(x)\times I.F.]dx+C.$$

• If an initial or boundary condition is given, substitute it into the general solution to determine the arbitrary constant.

If the equation is instead expressed with $x$ as the dependent variable, use

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

and calculate the integrating factor with respect to $y$.

5. Exact differential equations

• Write the equation in the form

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

• Calculate

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

• Check exactness using

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

• If the equation is exact, find a function whose total differential is

$$M(x,y)dx+N(x,y)dy.$$

• Set that function equal to a constant to obtain the solution.

Where It Goes Wrong

• The order is confused with the degree: the order is determined by the highest derivative, whereas the degree is the power of that derivative when the equation is polynomial in its derivatives.

• During formation, arbitrary constants are not fully eliminated. If a relation contains $n$ arbitrary constants, it must be differentiated $n$ times before elimination.

• In a homogeneous equation, the substitution $y=vx$ is incomplete unless

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

• Separation of variables is applied without placing all terms involving one variable on one side and all terms involving the other variable on the other side.

• For a linear equation, the integrating factor is calculated from $P(x)$ in the standard form

$$\frac{dy}{dx}+P(x)y=Q(x),$$ not from an equation that has not first been rearranged into that form.

• The constant of integration or the condition for determining it is omitted. A given initial or boundary condition selects a particular solution from the general solution.

What Gets Asked

• Define a differential equation, its order, and its degree.

• Determine whether a differential equation is first-order and first-degree.

• Form a differential equation by eliminating arbitrary constants, including the example

$$y=a{e}^x+b{e}^{-x}⟹\frac{d^{2}y}{d{x}^2}-y=0.$$

• Solve equations by separation of variables, including equations of the form

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

• Identify and solve a homogeneous equation using

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

• Convert and solve a first-order linear equation using

$$I.F.=e^{\int P(x)dx}$$ and $$y\times I.F.=\int [Q(x)\times I.F.]dx+C.$$

• Solve an equation written in the alternative linear form

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

• Test whether

$$M(x,y)dx+N(x,y)dy=0$$ is exact using $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x},$$ and obtain its solution if it is exact.

• Use an initial or boundary condition to obtain a particular solution from a general solution.

• Verify a solution by substitution or differentiation.