H: Integration: Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor).

The idea

This topic covers solving first-order differential equations where the variables can be separated. You rearrange the equation so that all terms involving y are on one side and all terms involving x on the other, then integrate both sides. For example, dy/dx = xy can be written as (1/y) dy = x dx, leading to ln|y| = x^2/2 + C. If an initial condition is given, substitute to find the particular solution. Sometimes factorisation is needed first, such as dy/dx = x(y+1) becoming (1/(y+1)) dy = x dx. The solution may be left in implicit form or rearranged to make y the subject. Assessment focuses on correct separation, integration, and use of initial conditions.

What earns the marks

• Separate the variables correctly, ensuring all y terms are on one side and all x terms on the other.
• Integrate both sides with respect to their respective variables, including the constant of integration.
• Apply the initial condition to determine the value of the constant and obtain the particular solution.
• Rearrange the solution to the required form, such as y = f(x), if specified.
• Factorise expressions where necessary to enable separation, e.g., common factors.

Common mistakes and how to put them right

• Forgetting to include the constant of integration after integrating.

  Put it right: Add C immediately after integration.

• Incorrectly separating variables, e.g., not dividing both sides by the function of y.

  Put it right: Ensure the equation is in the form f(y) dy = g(x) dx.

• Misapplying the initial condition, such as substituting into the wrong side.

  Put it right: Substitute the given x and y values into the integrated equation to solve for C.

• Not factorising before separating, leading to an inseparable equation.

  Put it right: Look for common factors and factorise first.

Exam tips

• Always separate variables before integrating; do not integrate terms that are multiplied together.
• When finding a particular solution, substitute the initial condition as soon as possible to find C.
• If the question asks for y in terms of x, rearrange carefully, especially when logarithms or exponentials are involved.
• Check your solution by differentiating to see if it satisfies the original differential equation.