Differential equation: dy/dx = f(x,y)
Separable variables: dy/f(y) = dx/g(x)
Homogeneous equation: dy/dx = f(x/y)
Linear equation: dy/dx + Py = Q
Bernoulli's equation: dy/dx + Py = Qyⁿ
Exact equation: ∂M/∂y = ∂N/∂x
Integrating factor: I.F = eⁱ∫Pdx
Solution of linear equation: y(I.F) = ∫Q(I.F)dx + c
Euler's method: yₙ₊₁ = yₙ + hf(xₙ,yₙ)
Runge-Kutta method: yₙ₊₁ = yₙ + (1/6)(k₁ + 2k₂ + 2k₃ + k₄)
Laplace transform: F(s) = ∫₀∞ e⁻sx f(x) dx
Inverse Laplace transform: f(x) = (1/2πi) ∫γ-ie⁺sx F(s) ds

🪤 The 5 Mistakes That Cost Marks

Not checking the differential equation for exactness
Forgetting to multiply by the integrating factor
Not using the correct method for solving the differential equation
Not applying the boundary conditions correctly
Not simplifying the final solution

✏️ 3 Solved PYQs

Solve the differential equation: dy/dx = (x+y)/(x-y) Step 1: Put x+y = v, so dx/dy = dv/dy - 1 Step 2: Substitute ∈ the differential equation: dv/dy - 1 = v/(v-2y) Step 3: Simplify and solve: v = 2y + ce²ᵞ Step 4: Substitute back x+y = v: x+y = 2y + ce²ᵞ Step 5: Simplify: x = y + ce²ᵞ
Solve the differential equation: (dy/dx) = (x²-y²)/(x+y) Step 1: Put x²-y² = u, so 2x-2y(dy/dx) = du/dx Step 2: Substitute ∈ the differential equation: 2x-2y(dy/dx) = (u)/(x+y) Step 3: Simplify and solve: 2x-2y(dy/dx) = (x²-y²)/(x+y) Step 4: Substitute back u = x²-y²: 2x-2y(dy/dx) = u/(x+y) Step 5: Simplify: 2x-2y(dy/dx) = (x²-y²)/(x+y)
Solve the differential equation: (dy/dx) + (y/x) = x³y Step 1: Put y = vx, so dy/dx = v + x(dv/dx) Step 2: Substitute ∈ the differential equation: v + x(dv/dx) + v = x³vx Step 3: Simplify and solve: x(dv/dx) = x³v - 2v Step 4: Separate variables: dv/v = (x³-2)dx/x Step 5: Integrate: ∫dv/v = ∫(x³-2)dx/x

🧠 The One Thing Most Students Get Wrong

Most students get wrong the application of the integrating factor ∈ linear differential equations
The integrating factor is eⁱ∫Pdx, where P is the coefficient of y ∈ the differential equation
For example, ∈ the differential equation dy/dx + 2y = x, the integrating factor is eⁱ∫2dx = e²ˣ
Multiplying the differential equation by the integrating factor, we get: e²ˣ(dy/dx) + 2e²ˣy = xe²ˣ
This can be written as: d/dx(ye²ˣ) = xe²ˣ
Integrating both sides, we get: ye²ˣ = ∫xe²ˣ dx + c

👁️ Ayush's Note

To solve differential equations quickly, use the following shortcuts:
- For separable variables, separate the variables and integrate
- For homogeneous equations, put y = vx and solve
- For linear equations, use the integrating factor
- For exact equations, use the test for exactness and solve
Also, use the following formulas:
- ∫eⁱˣ dx = eⁱˣ
- ∫e⁻ˣ dx = -e⁻ˣ
- ∫(1/x) dx = ln|x|
And, practice the following types of questions:
- Solving differential equations using various methods
- Finding the general solution and the particular solution
- Applying boundary conditions to find the particular solution

🔁 Last 5 Minutes Box

Revision of formulas: dy/dx = f(x,y), dy/f(y) = dx/g(x), etc.
Revision of methods: separable variables, homogeneous equation, linear equation, etc.
Revision of shortcuts: using integrating factor, using test for exactness, etc.
Practice of solving differential equations quickly
Practice of applying boundary conditions to find the particular solution

📝 Practice MCQs

1. The differential equation dy/dx = (x+y)/(x-y) has the solution

A) x+y = ce²ˣ

B) x-y = ce²ˣ

C) x+y = 2y + ce²ᵞ

D) x-y = 2y + ce²ᵞ

Answer: C) x+y = 2y + ce²ᵞ

2. The differential equation (dy/dx) = (x²-y²)/(x+y) has the solution

A) x²-y² = (x+y)² + c

B) x²-y² = (x+y)² - c

C) x²-y² = (x+y) + c

D) x²-y² = (x+y) - c

Answer: A) x²-y² = (x+y)² + c

3. The differential equation (dy/dx) + (y/x) = x³y has the solution

A) y = (x⁴/4 + c)/x

B) y = (x⁴/4 - c)/x

C) y = (x⁴/4 + c)x

D) y = (x⁴/4 - c)x

Answer: A) y = (x⁴/4 + c)/x

4. The differential equation dy/dx = (x+y)/(x-y) is

A) Homogeneous

B) Linear

C) Exact

D) Separable

Answer: A) Homogeneous

5. The differential equation (dy/dx) + (y/x) = x³y is

A) Homogeneous

B) Linear

C) Exact

D) Bernoulli's equation

Answer: B) Linear

🚀 Ready to Ace Your Exam?

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🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides:

Differential equation: dy/dx = f(x,y)
Separable variables: dy/f(y) = dx/g(x)
Homogeneous equation: dy/dx = f(x/y)
Linear equation: dy/dx + Py = Q
Bernoulli's equation: dy/dx + Py = Qyⁿ
Exact equation: ∂M/∂y = ∂N/∂x
Integrating factor: I.F = eⁱ∫Pdx
Solution of linear equation: y(I.F) = ∫Q(I.F)dx + c
Euler's method: yₙ₊₁ = yₙ + hf(xₙ,yₙ)
Runge-Kutta method: yₙ₊₁ = yₙ + (1/6)(k₁ + 2k₂ + 2k₃ + k₄)
Laplace transform: F(s) = ∫₀∞ e⁻sx f(x) dx
Inverse Laplace transform: f(x) = (1/2πi) ∫γ-ie⁺sx F(s) ds

🪤 The 5 Mistakes That Cost Marks

Not checking the differential equation for exactness
Forgetting to multiply by the integrating factor
Not using the correct method for solving the differential equation
Not applying the boundary conditions correctly
Not simplifying the final solution

✏️ 3 Solved PYQs

Solve the differential equation: dy/dx = (x+y)/(x-y) Step 1: Put x+y = v, so dx/dy = dv/dy - 1 Step 2: Substitute ∈ the differential equation: dv/dy - 1 = v/(v-2y) Step 3: Simplify and solve: v = 2y + ce²ᵞ Step 4: Substitute back x+y = v: x+y = 2y + ce²ᵞ Step 5: Simplify: x = y + ce²ᵞ
Solve the differential equation: (dy/dx) = (x²-y²)/(x+y) Step 1: Put x²-y² = u, so 2x-2y(dy/dx) = du/dx Step 2: Substitute ∈ the differential equation: 2x-2y(dy/dx) = (u)/(x+y) Step 3: Simplify and solve: 2x-2y(dy/dx) = (x²-y²)/(x+y) Step 4: Substitute back u = x²-y²: 2x-2y(dy/dx) = u/(x+y) Step 5: Simplify: 2x-2y(dy/dx) = (x²-y²)/(x+y)
Solve the differential equation: (dy/dx) + (y/x) = x³y Step 1: Put y = vx, so dy/dx = v + x(dv/dx) Step 2: Substitute ∈ the differential equation: v + x(dv/dx) + v = x³vx Step 3: Simplify and solve: x(dv/dx) = x³v - 2v Step 4: Separate variables: dv/v = (x³-2)dx/x Step 5: Integrate: ∫dv/v = ∫(x³-2)dx/x

🧠 The One Thing Most Students Get Wrong

Most students get wrong the application of the integrating factor ∈ linear differential equations
The integrating factor is eⁱ∫Pdx, where P is the coefficient of y ∈ the differential equation
For example, ∈ the differential equation dy/dx + 2y = x, the integrating factor is eⁱ∫2dx = e²ˣ
Multiplying the differential equation by the integrating factor, we get: e²ˣ(dy/dx) + 2e²ˣy = xe²ˣ
This can be written as: d/dx(ye²ˣ) = xe²ˣ
Integrating both sides, we get: ye²ˣ = ∫xe²ˣ dx + c

👁️ Ayush's Note

To solve differential equations quickly, use the following shortcuts:
- For separable variables, separate the variables and integrate
- For homogeneous equations, put y = vx and solve
- For linear equations, use the integrating factor
- For exact equations, use the test for exactness and solve
Also, use the following formulas:
- ∫eⁱˣ dx = eⁱˣ
- ∫e⁻ˣ dx = -e⁻ˣ
- ∫(1/x) dx = ln|x|
And, practice the following types of questions:
- Solving differential equations using various methods
- Finding the general solution and the particular solution
- Applying boundary conditions to find the particular solution

🔁 Last 5 Minutes Box

Revision of formulas: dy/dx = f(x,y), dy/f(y) = dx/g(x), etc.
Revision of methods: separable variables, homogeneous equation, linear equation, etc.
Revision of shortcuts: using integrating factor, using test for exactness, etc.
Practice of solving differential equations quickly
Practice of applying boundary conditions to find the particular solution

📝 Practice MCQs

1. The differential equation dy/dx = (x+y)/(x-y) has the solution

A) x+y = ce²ˣ

B) x-y = ce²ˣ

C) x+y = 2y + ce²ᵞ

D) x-y = 2y + ce²ᵞ

Answer: C) x+y = 2y + ce²ᵞ

2. The differential equation (dy/dx) = (x²-y²)/(x+y) has the solution

A) x²-y² = (x+y)² + c

B) x²-y² = (x+y)² - c

C) x²-y² = (x+y) + c

D) x²-y² = (x+y) - c

Answer: A) x²-y² = (x+y)² + c

3. The differential equation (dy/dx) + (y/x) = x³y has the solution

A) y = (x⁴/4 + c)/x

B) y = (x⁴/4 - c)/x

C) y = (x⁴/4 + c)x

D) y = (x⁴/4 - c)x

Answer: A) y = (x⁴/4 + c)/x

4. The differential equation dy/dx = (x+y)/(x-y) is

A) Homogeneous

B) Linear

C) Exact

D) Separable

Answer: A) Homogeneous

5. The differential equation (dy/dx) + (y/x) = x³y is

A) Homogeneous

B) Linear

C) Exact

D) Bernoulli's equation

Answer: B) Linear

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides: