Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Overview
An ordinary differential equation (ODE) relates a function with its derivatives. This chapter covers order/degree and three first-order solution methods.
1. Order and Degree
Order = highest derivative present
Degree = power of the highest-order derivative (equation cleared of radicals)
2. Solution Methods (first order)
Variable separable: dy/g(y) = f(x)dx → integrate both sides
Homogeneous: dy/dx = F(y/x) → put y = vx, dy/dx = v + x dv/dx
Linear: dy/dx + Py = Q → IF = e^∫P dx → y·IF = ∫(Q·IF)dx + c
Worked Examples
Example 1 — Order and degree
Find order and degree of (d²y/dx²)³ + (dy/dx)² + y = 0.
Highest derivative = d²y/dx² → order 2
Its power = 3 → degree 3
Example 2 — Variable separable
Solve dy/dx = xy.
dy/y = x dx → ∫dy/y = ∫x dx
ln|y| = x²/2 + c → y = A e^{x²/2}
Example 3 — Homogeneous
Solve dy/dx = (x + y)/x.
= 1 + y/x. Put y = vx → v + x dv/dx = 1 + v → x dv/dx = 1
dv = dx/x → v = ln|x| + c → y/x = ln|x| + c → y = x ln|x| + cx
Example 4 — Linear ODE
Solve dy/dx + y = e^x.
P = 1, Q = e^x → IF = e^∫1 dx = e^x
y·e^x = ∫ e^x · e^x dx = ∫ e^{2x} dx = e^{2x}/2 + c
y = e^x/2 + c e^{−x}
Common Mistakes
- Reading degree before clearing radicals/fractions in derivatives.
- Forgetting the constant of integration.
- Wrong integrating factor (IF = e^∫P dx, not e^P).
- Not substituting back y = vx after solving the homogeneous form.
Exam Tips & Revision Checklist
- Identify the type (separable / homogeneous / linear) before solving.
- Always add + c and simplify.
- Practised: order/degree, variable separable, homogeneous, linear ODE with IF.
Frequently Asked Questions
Order = highest derivative present; degree = power of the highest-order derivative after the equation is made free of radicals/fractions in derivatives.