Differential Equations Tutorial

Differential Equations Tutorial

1. Order and Degree

State the order and degree of the following differential equations:

  • (a) $$5 \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + (1 - x) y = \sin y$$

    • Order: 2
    • Degree: 1

  • (b) $$5(y'') - y = e^x$$

    • Order: 2
    • Degree: 1

  • (c) $$\frac{d^2 y}{dx^2} + x\left(\frac{dy}{dx}\right) + y = 0$$

    • Order: 2
    • Degree: 1

  • (d) $$y' + y^2 x = 2x^3$$

    • Order: 1
    • Degree: 1

2. Separable or Non-Separable

State whether each differential equation is separable ($g(y) dy = f(x) dx$) or non-separable.

  • (a) $$\frac{dy}{dx} + x^2 y = x$$

    • Non-separable

  • (b) $$y' - x^2 y^2 = x^2$$

    • Separable

  • (c) $$\frac{dy}{dx} = -\frac{x}{y-3}$$

    • Separable

  • (d) $$\frac{dy}{dx} - 2xy = x^2 - x$$

    • Non-separable

3. General Solution

Find the general solution for each differential equation.

  • (a) $$\frac{dy}{dx} - x \sqrt{1+x^2} = 0$$

    • $y = \frac{1}{3} (1+x^2)^{3/2} + C$

  • (b) $$e^y \frac{dy}{dx} + \sin x = 0$$

    • $e^y = \cos x + C$

  • (c) $$(x+1) \frac{dy}{dx} = x(y+3)$$

    • $y+3 = C (x+1) e^{x}$

  • (d) $$3y^2 \frac{dy}{dx} + 2x = 1$$

    • $y^3 = x - x^2 + C$

  • (e) $$x , dy - y , dx = 0$$

    • $y = Cx$

  • (f) $$3y , dx + (xy + 5x) , dy = 0$$

    • $3 \ln|y| + y + 5 \ln|y| = - \ln|x| + C$
    • Simplified: $8 \ln|y| + y = - \ln|x| + C$

  • (g) $$(y + yx^2) , dy + (x + xy^2) , dx = 0$$

    • $\frac{1}{2} \ln|1+y^2| + \frac{1}{2} \ln|1+x^2| = C$

4. Particular Solution

Find the particular solution for each differential equation with the given initial condition.

  • (a) $$\frac{dy}{dx} = 1 - \frac{2}{y}, \quad y=3 \text{ when } x=3$$

    • $y + 2 \ln|y-2| = x + C$
    • Particular: $y + 2 \ln|y-2| = x + 2 \ln 1 = x$

  • (b) $$(xy^2 - xy) , dx - 2 , dy = 0, \quad y=2 \text{ when } x=0$$

    • $x - \ln|x| - \frac{2}{y} = C$
    • Particular: $x - \ln|x| - \frac{2}{y} = -1$

  • (c) $$\cos y , dx + x \sin y , dy = 0, \quad y(3) = \frac{\pi}{3}$$

    • $\ln|x| + \ln|\sin y| = C$
    • Particular: $\ln|x| + \ln|\sin y| = \ln 3 + \ln\left(\frac{\sqrt{3}}{2}\right)$

5. Verification of Solution

Prove that $y$ is a solution of the differential equation by implicit differentiation.

  • (a) $$(x - 2y) \frac{dy}{dx} + 2x + y = 0, \quad y^2 - x^2 - xy = C$$

    • Differentiate implicitly: $2y y' - 2x - y - x y' = 0$
    • Simplify: $(2y - x) y' = 2x + y$
    • Multiply both sides by $-1$: $(x-2y) y' + 2x + y = 0$
    • Verified.

  • (b) $$y \frac{dy}{dx} = x, \quad x^2 - y^2 = C$$

    • Differentiate implicitly: $2x - 2y y' = 0$
    • Simplify: $x - y y' = 0$
    • Rearranged: $y y' = x$
    • Verified.

Source: Tutorial 7 FAD1014 2025_2026.pdf