Chapter 9: Differential Equations

Key Concepts:
• Order: Highest order derivative present
• Degree: Power of highest order derivative
• Variable Separable: Separate x and y terms
• Linear DE: dy/dx + Py = Q

Q1. Find the order and degree of: d²y/dx² + (dy/dx)³ + y = 0.

Solution

Highest order derivative = d²y/dx² (order = 2)
Power of highest order derivative = 1 (degree = 1)
Order = 2, Degree = 1

Q2. Solve: dy/dx = (1 + y²)/(1 + x²)

Solution

dy/(1 + y²) = dx/(1 + x²)

Integrating both sides:
tan⁻¹y = tan⁻¹x + C

General Solution: tan⁻¹y − tan⁻¹x = C

Q3. Solve: dy/dx + y = e^x

Solution

This is a linear differential equation of the form dy/dx + Py = Q where P = 1, Q = e^x.

I.F. = e^{∫ 1 dx} = e^x

y · e^x = ∫ e^x · e^x dx = ∫ e^2x dx
y · e^x = e^2x/2 + C

y = e^x/2 + Ce^−x

Q4. Solve: x dy/dx + 2y = x² log x

Solution

dy/dx + (2/x)y = x log x (dividing by x)

P = 2/x, Q = x log x
I.F. = e^{∫ (2/x) dx} = e^{2 log x} = x²

y · x² = ∫ x² · x log x dx = ∫ x³ log x dx

Using integration by parts for ∫ x³ log x dx:
= log x · x&sup4;/4 − ∫ (x&sup4;/4)(1/x) dx
= x&sup4; log x/4 − x&sup4;/16

y = x² log x/4 − x²/16 + Cx⁻²

Q5. Solve: y dx + (x − y²) dy = 0

Solution

dx/dy = (−x + y²)/y = −x/y + y
dx/dy + x/y = y (linear in x)

I.F. = e^{∫ (1/y) dy} = y

x · y = ∫ y · y dy = ∫ y² dy = y³/3 + C

xy = y³/3 + C

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