Exact Differential Equation Trick in 59 Seconds! 🔥 | JEE Maths

 

❓ Concept

🎬 Exact DE Trick

JEE mein jab differential equation dikhe:

$$M(x,y)dy+N(x,y)dx=0$$

sabse pehla sawaal hona chahiye 👇
👉 “Kya yeh exact hai?”
Agar exact nikla — solution almost direct mil jaata hai.

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🖼️ Concept Image

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✍️ Short Explanation

Exact DE questions JEE ke sabse scoring problems hote hain.
Reason simple hai:

👉 Na substitution
👉 Na heavy algebra
👉 Sirf partial derivatives + integration

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🔹 Step 1 — Form of an Exact Differential Equation

Given form:

$$M(x,y)\,dy + N(x,y)\,dx = 0$$

✅ Exactness Condition:

$$\frac{\mathrm{\partial }M}{\mathrm{\partial }x}=\frac{\mathrm{\partial }N}{\mathrm{\partial }y}$$

📌 Agar dono equal aa jaayein:

• DE exact hai

• Direct integration allowed

Agar equal nahi:

• Phir integrating factor ka case hota hai (alag topic)

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🔹 Step 2 — Meaning of Exactness

Exact hone ka matlab 👇

Ek function $F(x,y)$ exist karta hai aisa ki:

$$dF = M\,dy + N\,dx$$

Isliye DE ka general solution hota hai:

$$F(x,y) = C$$

👉 Poora problem isi $F(x,y)$ ko find karna hota hai.

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🔹 Step 3 — How to Solve an Exact DE (CORE STEPS 🔥)

Follow this fixed 5-step method:

1️⃣ Integrate $N(x,y)$ w.r.t. $x$

$$\int N(x,y)\,dx$$

2️⃣ Add a function of $y$

$$+\,g(y)$$

3️⃣ Differentiate this result w.r.t. $y$

4️⃣ Match with $M(x,y)$
→ Isse $g'(y)$ mil jaata hai

5️⃣ Integrate $g'(y)$
→ Final $F(x,y)$ mil jaata hai

Finally write:

$$F(x,y)=C$$

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🔹 Step 4 — Using the Initial Condition

Agar question deta hai:

$$(x_0,\ y_0)$$

Toh:

$$F(x_0,y_0) = C$$

👉 Isse constant $C$ milta hai
👉 Solution particular solution ban jaata hai

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🔹 Step 5 — To Find $y(a)$

Agar question bole:

Find $y(a)$

Then:
1️⃣ Final equation $F(x,y)=C$ lo
2️⃣ $x=a$ put karo
3️⃣ Equation solve karo
4️⃣ Required value mil jaata hai: $y(a)$

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✅ Final Takeaway

🧠 Exact DE Golden Rules:

• Form dekho: $Mdy + Ndx = 0$
  

• Exactness test: $\partial M/\partial x = \partial N/\partial y$
  

• Exact hai → direct integration

• Initial condition → constant

• Plug value → final answer

Agar yeh click ho gaya,
👉 DE questions JEE mein free marks ban jaate hain.

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