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Learn Module I: Partial Differential Equations – origin, Lagrange’s equation, Charpit’s method, Cauchy’s characteristics & higher-order PDEs with formulas.
Module II: Applications of Partial Differential Equations – classification, separation of variables, wave, heat, Laplace equations & transmission lines.
Module III: Statistical Techniques I – measures of central tendency, moments, MGF, skewness, kurtosis, curve fitting, correlation & regression analysis.
Module IV: Probability & Distributions – laws of probability, Bayes’ theorem, random variables, expectation, binomial, Poisson & normal distributions.
Module V: Sampling, Hypothesis Testing & Statistical Quality Control – t-test, F-test, Chi-square, ANOVA, X̄-R charts, p-np-c control charts with formulas.
Practice questions with detailed solutions for Module I: Partial Differential Equations – Lagrange, Charpit, Cauchy, higher-order PDEs & reducible equations.
Practice questions with solutions for Module II: Applications of PDEs – classification, separation of variables, wave, heat, Laplace equations & transmission lines.
Practice questions with solutions for Module III: Statistical Techniques I – central tendency, moments, MGF, curve fitting, correlation & regression analysis.
Practice questions with solutions for Module IV: Probability & Distributions – Bayes’ theorem, random variables, binomial, Poisson & normal distributions.
Practice questions with solutions for Module V: Sampling, Hypothesis Testing & SQC – t-test, F-test, Chi-square, ANOVA, X̄-R charts, p-np-c control charts.
Complete integration formulas from basics to intermediate level. Covers substitution, parts, partial fractions, trigonometric integrals, definite integrals, and solved examples.
Complete differentiation formulas from basics to intermediate level. Covers power rule, chain rule, product rule, quotient rule, trigonometric, exponential, logarithmic, implicit, parametric, and solved examples.
Learn PDEs, probability, statistics, MGF, regression, hypothesis testing, ANOVA, and statistical quality control with formulas and examples.
Level
intermediate
Estimated Hours
30 hrs
Total Formulas
0
Total Lessons
0
Origin of PDEs
Partial differential equations arise when a quantity depends on more than one independent variable (e.g., temperature depends on space and time).
Linear & Non-linear First-order PDEs
A general first-order PDE is
Lagrange’s Linear Equation
Solution by solving the subsidiary equations
Charpit’s Method (for non-linear first-order PDEs)
Auxiliary equations:
Cauchy’s Method of Characteristics
Treat the first-order PDE as a system of ODEs along characteristic curves.
Higher-order Linear PDEs with Constant Coefficients
Auxiliary equation obtained by substituting .
Equations Reducible to Linear Form with Constant Coefficients
Change of independent variables that transforms variable-coefficient equations into constant-coefficient ones.
Classification of Second-order Linear PDEs
Discriminant :
| Type | Canonical Form | |
|---|---|---|
| Hyperbolic | ||
| Parabolic | ||
| Elliptic |
Method of Separation of Variables
Assume (or ) and reduce the PDE to ordinary differential equations.
Wave Equation (up to two dimensions)
1-D:
2-D:
Heat Conduction Equation
1-D and 2-D forms are obtained similarly.
Laplace Equation in Two Dimensions
(or in polar coordinates ).
Transmission-Line Equations
Telegraph equations derived from Kirchhoff’s laws, reducing to wave or heat-type equations under special conditions.
Measures of Central Tendency
Mean, Median, Mode, Geometric Mean, Harmonic Mean.
Moments
Raw moment about origin:
Central moment:
Moment Generating Function (MGF)
Skewness & Kurtosis
Curve Fitting – Method of Least Squares
Minimize .
Straight line:
Normal equations:
Second-degree parabola:
Exponential curves: or (take logarithm).
Correlation
Pearson’s coefficient:
Spearman’s rank correlation:
Regression
Lines of regression:
where
Properties: , etc.
Non-linear regression obtained by suitable transformations or higher-degree polynomials.
Probability
Random Variables
Expectation & Variance
Important Distributions
| Distribution | pmf / pdf | Mean | Variance |
|---|---|---|---|
| Binomial | |||
| Poisson | |||
| Normal |
Sampling Theory
Hypothesis Testing
Important Tests
-test (mean, difference of means, paired)
-test (equality of two variances)
Chi-square test (goodness of fit, independence, variance)
One-way ANOVA
Partition total sum of squares:
Test statistic .
Statistical Quality Control (SQC)
Control charts monitor process stability.
Variable Charts
Attribute Charts
Control limits are typically set at (3-sigma limits).
Name: Ankit kushwaha
Email: ankitkushwaha909@gmail.com
SEO Title: Engineering Mathematics: PDE & Statistical Techniques
SEO Description: Learn PDEs, probability, statistics, MGF, regression, hypothesis testing, ANOVA, and statistical quality control with formulas and examples.
Created: 8/4/2026, 7:42:25 AM
Updated: 8/12/2026, 8:59:51 AM