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(a) Solve the partial differential equation (D² − 3DDʹ)z = 0.

The auxiliary equation is obtained by replacing D with m and Dʹ with 1:

m² − 3m = 0 ⇒ m(m − 3) = 0 ⇒ m = 0, 3

Solution: z = f₁(y) + f₂(y + 3x)

(b) Classify the PDE: 4∂²u/∂x² − 4∂²u/∂x∂t + ∂²u/∂t² = 0

 Comparing with A·u_xx + B·u_xt + C·u_tt = 0: A = 4, B = −4, C = 1

Discriminant = B² − 4AC = 16 − 16 = 0

Since B² − 4AC = 0, the equation is Parabolic.

(c) Write the formula for Rank

## **UNIT-I: Word Formation and Roots**
### **The Process of Word Formation**
English expands its vocabulary through several mechanical processes:
 * **Derivation:** This involves adding **affixes** (prefixes or suffixes) to a root word to change its meaning or grammatical category.
   * *Example:* Adding "-ness" to the adjective "happy" creates the noun "happiness."
 * **Compounding:** Joining two or more complete words to create a new concept.
   * *Example:* "Sun" + "flower" = "Sunflower"; "Notebook.

 1. ਵਾਕ ਰਚਨਾ (Sentence Structure)
ਵਾਕ ਭਾਸ਼ਾ ਦੀ ਸਭ ਤੋਂ ਵੱਡੀ ਸੁਤੰਤਰ ਇਕਾਈ ਹੈ। ਵਾਕ ਰਚਨਾ ਦੇ ਮੁੱਖ ਦੋ ਭਾਗ ਹੁੰਦੇ ਹਨ:       *ਉਦੇਸ਼ (Subject):** ਵਾਕ ਵਿੱਚ ਜਿਸ ਬਾਰੇ ਗੱਲ ਕੀਤੀ ਜਾਵੇ।
 * **ਵਿਧੇ (Predicate):** ਉਦੇਸ਼ ਬਾਰੇ ਜੋ ਕੁਝ ਕਿਹਾ ਜਾਵੇ।
**ਬਣਤਰ ਦੇ ਆਧਾਰ

### 1. ਸ਼ਬਦ ਰਚਨਾ (Word Formation)
ਸ਼ਬਦ ਰਚਨਾ ਦਾ ਅਰਥ ਹੈ—ਵੱਖ-ਵੱਖ ਵਿਧੀਆਂ ਰਾਹੀਂ ਨਵੇਂ ਸ਼ਬਦਾਂ ਦਾ ਨਿਰਮਾਣ ਕਰਨਾ। ਪੰਜਾਬੀ ਵਿੱਚ ਬਣਤਰ ਦੇ ਆਧਾਰ 'ਤੇ ਸ਼ਬਦ ਦੋ ਪ੍ਰਕਾਰ ਦੇ ਹੁੰਦੇ ਹਨ:
 * **ਮੂਲ ਸ਼ਬਦ:** ਜਿਹੜੇ ਸ਼ਬਦ ਆਪਣੇ ਆਪ ਵਿੱਚ ਪੂਰਨ ਹੁੰਦੇ ਹਨ

Classification of Violent Actors-

• Warlords: These are individuals with military experience who rely on force and violence to achieve their goals. They are skilled in managing armed groups and profiting from ongoing conflicts and disputes.
• Militias: These are irregular armed groups that exploit the weakness of the state and seek to acquire resources, money, power, and control. They may represent ethnic, religious, or sectarian groups, as seen in Iraq, Syria, and Libya.
• Paramilitary Forces and

My name is Peter, and I am a young student from Serbia. I live in Novi Sad, which is one of the biggest and most beautiful cities in the country. I am currently studying information technology, because computers and technology have always interested me. I enjoy learning how programs work and how software can solve everyday problems. In my opinion, technology is one of the most important parts of modern life, and I would like to work in this field in the future.

I consider myself to be a friendly,