Standard content

“Differential Equations in Depth: Substitutions, Exact Equations and Second Order” is available to Standard subscribers.

Beyond the textbook. Turn an equation that is not standard into one that is, by a substitution. Solve Bernoulli equations, exact equations and equations of the form $\dfrac{dy}{dx} = f(ax + by + c)$. Then meet the second-order equation with constant coefficients, as used in A-Level Further Maths. It is at the level of JEE Advanced, A-Level Further Maths and IB Higher Level, with a full timed paper.

3Lessons12Problems2Python Labs1Question Set1Mock Test

Read the first lesson freeSubstitutions and Bernoulli equations

What's inside

Substitutions and Bernoulli equations: Many equations are not separable, homogeneous or linear as they stand, but a substitution makes them so. The equation $\dfrac{dy}{dx} = f(ax + by + c)$

The chapter itself stays free

This is an extra course for Differential Equations. Its main courses are open to everyone:

Create an account

Exact equations: An equation written as

M\,dx + N\,dy = 0

is exact when

M\,dx + N\,dy

is the total change of one function

F(x, y)

. The test is

\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}

, and the solution is

F(x, y) = C

. When the test fails, a suitable integrating factor often makes the equation exact. Inspection of familiar groups such as

x\,dy + y\,dx

gives a quick solution.

Second-order equations with constant coefficients: A mass on a spring, a swinging door and a circuit with a coil and a capacitor all obey a second-order equation

ay'' + by' + cy = f(x)

with constant coefficients. Try

y = e^{mx}

in the equation with

f = 0

and you reach the auxiliary equation

am^2 + bm + c = 0

. Its roots, real, equal or complex, decide the form of the solution. A particular integral takes care of

f(x)