11.10PureStretch

Solving Differential Equations

A first-order differential equation can be solved by separating the variables — getting all the y-terms on one side and the x-terms on the other — then integrating both sides.

28 min Video by Zeeshan Zamurred Integration

Edexcel A level Maths: 11.10 Solving Differential Equations (Part 1)Watch the full walkthrough before the notes below.

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What you'll be able to do

Separate variables in a differential equation
Integrate both sides
Find the general solution (with + c)
Use a boundary condition for the particular solution

Separating variables

For $\frac{d y}{d x} = f (x) g (y)$ , rearrange to $\frac{1}{g ( y )} d y = f (x) d x$ and integrate both sides. One constant of integration covers both.

\int \frac{1}{g ( y )} d y = \int f (x) d x

Separation of variables.

\int \frac{1}{y} d y = \int x d x

ln ∣ y ∣ = \frac{x ^{2}}{2} + c

, so

y = A e^{x^{2} /2}

Answer

y = A e^{x^{2} /2}

Tip — The general solution keeps an arbitrary constant; a boundary condition pins it down.

General vs particular

The $general solution$ contains the arbitrary constant. Substituting a given condition (e.g. $y = 1$ when $x = 0$ ) gives the $particular solution$ .

Formula recap

\int \frac{1}{g ( y )} d y = \int f (x) d x

Separate and integrate.

y = A e^{k x}

Typical exponential solution.

Common mistakes to avoid

Forgetting the constant of integration.

The general solution needs +c (or A).

Not using the boundary condition when asked for the particular solution.

Substitute the given values to find the constant.

Key takeaways

Separate variables: y-terms one side, x-terms the other.
Integrate both sides (one constant).
Use a boundary condition for the particular solution.

Test yourself

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