Chapters

  • Chapter 9
  • Calculus
  • 23 questions

Differential Equations

Order and degree, general and particular solutions, variable separable, homogeneous and linear differential equations.

A differential equation describes a situation by stating how a quantity changes rather than what it is. Solving one means recovering the quantity itself from that description.

The course covers three methods, and each is defined by the form of the equation, so the first step is always classification: can the variables be separated, is the equation homogeneous, or is it linear? Identifying the type correctly is most of the work; the integration that follows is familiar from earlier chapters.

What you should be able to do

  • State the order and degree of a differential equation, and recognise when the degree is undefined.
  • Distinguish general from particular solutions and apply initial conditions.
  • Solve differential equations by separating the variables.
  • Recognise and solve homogeneous differential equations of the first order and first degree.
  • Solve linear differential equations using an integrating factor.

Topics

5 topics

  1. Basic Concepts: Order and Degree

    Definition of a differential equation, and determining its order and degree.

  2. General and Particular Solutions

    The role of arbitrary constants, and using given conditions to find a particular solution.

  3. Variable Separable Method

    Separating variables and integrating both sides.

  4. Homogeneous Differential Equations

    Recognising homogeneity and using the substitution.

  5. Linear Differential Equations

    Equations of the form and, solved with an integrating factor.

Formulas

8 items · All formulas

  • Order of a differential equation

    \text{the order of the highest derivative present}

  • Degree of a differential equation

    \text{the power of the highest-order derivative, once the equation is a polynomial in its derivatives}
    Valid when
    Undefined if the equation cannot be written as a polynomial in the derivatives — for example if a derivative appears inside a sine, a logarithm or a radical.

  • Variable separable form

    \int \frac{dy}{g(y)} = \int f(x)\,dx + C
    Valid when
    Applies when \dfrac{dy}{dx} = f(x)g(y).

  • Homogeneous substitution

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

    Note the product rule — both terms are needed.

  • Linear differential equation in $y$

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

  • Integrating factor

    \text{I.F.} = e^{\int P\,dx}

    No constant of integration is included when forming the integrating factor.

  • Solution of a linear differential equation

    y \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.})\,dx + C

  • Linear in $x$ instead

    \frac{dx}{dy} + P(y)\,x = Q(y) \;\Longrightarrow\; \text{I.F.} = e^{\int P\,dy}

    Use when the equation is linear in x but not in y.

Key points

7 items · All key points

  • Classify before solving: separable, homogeneous, or linear. The form dictates the method.

  • Degree is undefined when a derivative sits inside a transcendental function or a radical.

    \sin\!\left(\frac{dy}{dx}\right) + y = 0 has order 1 but no degree.

  • In the substitution y = vx, the product rule gives \dfrac{dy}{dx} = v + x\dfrac{dv}{dx}.

  • The equation must be in the exact form \dfrac{dy}{dx} + Py = Q before you read off P.

    If the coefficient of \frac{dy}{dx} is not 1, divide through first.

  • Do not add a constant when computing the integrating factor.

  • A general solution of an nth-order equation contains exactly n arbitrary constants; a particular solution has none.

  • After solving a homogeneous equation, substitute v = \dfrac{y}{x} back — the answer must be in x and y.

Common mistakes

6 items · All common mistakes

  • Mistake

    Writing \dfrac{dy}{dx} = x\dfrac{dv}{dx} after substituting y = vx.

    Instead

    \dfrac{dy}{dx} = v + x\dfrac{dv}{dx} — y = vx is a product.

    Why

    The substitution is remembered as a formula rather than derived, so the product rule is skipped.

  • Mistake

    Reading off P(x) before dividing through by the coefficient of \frac{dy}{dx}.

    Instead

    Put the equation into the standard form \frac{dy}{dx} + Py = Q first.

    Why

    The equation often looks close enough to standard form that the division is judged unnecessary.

  • Mistake

    Including a constant of integration in the integrating factor.

    Instead

    Use \text{I.F.} = e^{\int P\,dx} with no constant; the constant enters once, at the final integration.

  • Mistake

    Leaving the answer to a homogeneous equation in terms of v.

    Instead

    Substitute v = \frac{y}{x} back to express the solution in x and y.

  • Mistake

    Assigning a degree to an equation in which a derivative appears inside a sine or a square root.

    Instead

    State that the degree is not defined. Order is still defined.

  • Mistake

    Omitting the constant of integration and so producing a particular solution when a general one was asked for.

    Instead

    Include +C; apply an initial condition only when the question supplies one.

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