Solving Simple Differential Equations
This topic covers how to reverse the process of differentiation. Given a function for the gradient of a curve (dy/dx), you will use integration to find the equation of the original curve (y).
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Solving dy/dx = f(x) means finding y by integrating f(x) with respect to x.
- The initial result of this integration is a 'general solution', which is a family of curves represented by y = F(x) + C, where F(x) is the integral of f(x) and C is an unknown constant.
- To find a unique 'particular solution', you need an 'initial condition' – a specific point (x, y) that the curve passes through.
- By substituting the x and y values from the initial condition into the general solution, you can solve for the constant C.
Diagram
Formulae
If dy/dx = f(x), then y = ∫f(x) dx Use this fundamental relationship when you are given the gradient function of a curve and need to find the equation of the curve itself.
Definitions
- Differential Equation
- An equation that relates a function with its derivatives. The form dy/dx = f(x) is a simple type of differential equation.
- General Solution
- The solution to a differential equation that includes an arbitrary constant of integration (+ C). It represents an entire family of functions.
- Particular Solution
- The specific solution obtained after using a given condition (like a point on the curve) to determine the value of the constant of integration, C.
Worked example
The gradient of a curve is given by the equation dy/dx = 6x2 + 8/x3. Given that the curve passes through the point (2, 19), find the equation for y.
- 1
First, rewrite the gradient function using negative indices for easier integration:
dy/dx = 6x2 + 8x-3 - 2
Integrate the function with respect to x to find the general solution for y:
y = ∫(6x2 + 8x-3) dx - 3
Apply the power rule for integration (∫axn dx = a(x^(n+1))/(n+1)):
y = (6x3)/3 + (8x-2)/-2 + C - 4
Simplify the expression:
y = 2x3 - 4x-2 + C - 5
Use the given point (2, 19) to find the constant C.
Substitute x=2 and y=19 into the equation - 6 19 = 2(2)3 - 4(2)-2 + C
- 7
Calculate the powers:
19 = 2(8) - 4(1/4) + C - 8
Simplify the arithmetic:
19 = 16 - 1 + C, which gives 19 = 15 + C - 9
Solve for C:
C = 4 - 10
Substitute C back into the general solution to get the particular solution:
y = 2x3 - 4x-2 + 4
Answer: y = 2x3 - 4/x2 + 4
Common mistakes
- ×Forgetting to include the constant of integration, '+ C', after integrating. This is the most common error.
- ×Making arithmetic mistakes when substituting coordinates to find C, especially with negative numbers or fractions.
- ×Incorrectly applying the power rule for integration to negative powers, for example, thinking the integral of x-3 is proportional to x-4 instead of x-2.
No-calculator tips
- ✓When finding C, calculate the numerical part of the equation first, then rearrange to find C. For example, in `19 = 2(8) - 4(1/4) + C`, first calculate `16 - 1 = 15` to get `19 = 15 + C`.
- ✓Always rewrite terms like `k/xn` as `kx-n` before integrating. It makes applying the power rule much more straightforward and less error-prone.
- ✓As a final check, mentally differentiate your final answer for y. It should return you to the original expression for dy/dx. This verifies your integration was correct.