Multivariable Calculus and First-Order Differential Equations

Course Information

• Institution: University of Mosul

• Department: Medical Physics Department

• Academic Year / Class: Class 2nd Year

• Subject: Calculus and Differential Equations

• Instructor: Dr. Akram Al-Abood

Multivariable Functions

• Definition: A function of more than one independent variable is a mathematical rule whose output values are determined by specifying the values of two or more independent variables.

• Functions of Two Variables: Expressed as $z = f(x, y)$, where $x$ and $y$ are independent variables, and $z$ is the dependent variable.

• Example 1: Right Circular Cylinder Volume

  • The volume $V$ depends on two independent variables: the radius $r$ and the height $h$

  • Formula: $V = f(r, h) = \text{π} \, r^2 \, h$

• Example 2: Right Circular Cylinder Surface Area

  • The surface area $S$ depends on the radius $r$ and height $h$

  • Formula: $S = f(r, h) = 2\text{π} \, r \, h + 2\text{π} \, r^2$

Partial Differentiation

• Definition: The partial derivative of a function containing two or more independent variables is defined as the rate of change of the function with respect to one specified independent variable, while holding all other independent variables constant.

• Scope: Analysis is focused on partial differentiation of functions of two independent variables $z = f(x, y)$.

• Notation:

  • Partial derivative with respect to $x$: $f_x(x, y) = \frac{\partial f}{\partial x} = \frac{\partial z}{\partial x}$

  • Partial derivative with respect to $y$: $f_y(x, y) = \frac{\partial f}{\partial y} = \frac{\partial z}{\partial y}$

• Worked Examples:

  • Problem 1: Given $f(x, y) = 2x^3 - 3x^2 \, y + y^3$

  • Derivative with respect to $x$: $f_x(x, y) = 6x^2 - 6x \, y$

  • Derivative with respect to $y$: $f_y(x, y) = -3x^2 + 3y^2$

  • Problem 2: Given $f(x, y) = -x^2 \, \sin(y) + x \, \cos(y^2)$

  • Derivative with respect to $x$: $f_x(x, y) = -2x \, \sin(y) + \cos(y^2)$

  • Derivative with respect to $y$: $f_y(x, y) = -x^2 \, \cos(y) - 2x \, y \, \sin(y^2)$

  • Problem 3: Given $f(x, y) = 3\sin^2(x) \, \sin(y) - 2\sin(x) \, \cos(y)$

  • Derivative with respect to $x$: $f_x(x, y) = 6\sin(x) \, \cos(x) \, \sin(y) - 2\cos(x) \, \cos(y)$

  • Derivative with respect to $y$: $f_y(x, y) = 3\sin^2(x) \, \cos(y) + 2\sin(x) \, \sin(y)$

  • Problem 4: Partial derivative with respect to $x$

  • Formula: $f_x(x, y) = -2e^{-2x} \, \sin(y) + \sin(x)$

  • Problem 5: Partial derivative with respect to $x$

  • Formula: $f_x(x, y) = -x \, \sec(x) \, \tan(x) + \sec(x)$

  • Problem 6: Given $z = 2x - 5y + e^{\cos(y)} + e^{\sin(2x)}$

  • Partial derivative with respect to $x$: $z_x = 2 + 2e^{\sin(2x)} \, \cos(2x)$

Differential Equations

• Definition: A differential equation is an equation involving independent variables, a dependent variable, and one or more derivatives or differentials of the dependent variable.

• Order: The order of a differential equation is defined as the order of the highest derivative present in the equation.

• Degree: The degree of a differential equation is the power (exponent) of the highest-order derivative, after the equation has been expressed in polynomial form regarding its derivatives.

• Example of Order and Degree:

  • Differential equation: $\left(\frac{d^3 y}{dx^3}\right)^2 + \left(\frac{d^2 y}{dx^2}\right)^5 + \frac{dy}{dx} = x$

  • Classification: Ordinary differential equation of Order 3 and Degree 2.

• First-Order Ordinary Differential Equations (ODEs):

  • Standard functional form of a 1st order, 1st degree ODE: $f\left(x, y, \frac{dy}{dx}\right) = 0$

  • Differential form: $M(x, y) \, dx + N(x, y) \, dy = 0$

• Definition of Solution:

  • A function $y = f(x)$ is defined as a solution of a differential equation if the function and its derivatives exist and satisfy the differential equation identically upon substitution.

• Solution Verification Example:

  • Problem: Show that $y = x \, \sin(x)$ is a solution of the differential equation: $\frac{d^2 y}{dx^2} + y = 2\cos(x)$

  • Step 1: First derivative using product rule:     $\frac{dy}{dx} = \frac{d}{dx}\left(x \, \sin(x)\right) = x \, \cos(x) + \sin(x)$

  • Step 2: Second derivative:     $\frac{d^2 y}{dx^2} = \frac{d}{dx}\left(x \, \cos(x) + \sin(x)\right) = -x \, \sin(x) + \cos(x) + \cos(x) = -x \, \sin(x) + 2\cos(x)$

  • Step 3: Substitution into Left-Hand Side (L.H.S.):     $\text{L.H.S.} = \frac{d^2 y}{dx^2} + y = -x \, \sin(x) + 2\cos(x) + x \, \sin(x) = 2\cos(x)$

  • Step 4: Comparison:     $\text{L.H.S.} = \text{R.H.S.} = 2\cos(x)$

  • Conclusion: $y = x \, \sin(x)$ is confirmed as a valid solution.

• Separation of Variables Method:

  • Applies to first-order differential equations of the form: $\frac{dy}{dx} = \frac{g(x)}{f(y)}$

  • Separation step: $f(y) \, dy = g(x) \, dx$

  • General solution by integration: $\int f(y) \, dy = \int g(x) \, dx + C$

Standard Integration Formulas

• Indefinite Integrals of Trigonometric Functions:

  1. $\int \sin(u) \, du = -\cos(u) + C$

  2. $\int \cos(u) \, du = \sin(u) + C$

  3. $\int \tan(u) \, du = -\ln\left|\cos(u)\right| + C = \ln\left|\sec(u)\right| + C$

  4. $\int \cot(u) \, du = \ln\left|\sin(u)\right| + C$

  5. $\int \sec(u) \, du = \ln\left|\sec(u) + \tan(u)\right| + C$

  6. $\int \csc(u) \, du = -\ln\left|\csc(u) + \cot(u)\right| + C$

  7. $\int \sec^2(u) \, du = \tan(u) + C$

  8. $\int \csc^2(u) \, du = -\cot(u) + C$

  9. $\int \sec(u) \, \tan(u) \, du = \sec(u) + C$

  10. $\int \csc(u) \, \cot(u) \, du = -\csc(u) + C$