The PDF document contains an assignment booklet for a course on Differential Equations (BMTC-132). Below is a breakdown of the assignment problems along with solutions for each:

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1. True or False Statements (10 × 2 = 20 Marks)

For each statement, we need to determine if it's true or false and provide a justification.

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2. Solve the Differential Equations

(a) Solve the equation:

$\frac{dy}{dx} = \frac{(1 + x)\tan(y)}{(1 + x)\sec(y) + e^x}$

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(b) Show that the following equation is integrable and find its integral:

$0 = x^2 + y^2 + z^2 + dx \cdot yz + dy \cdot zx + dz \cdot xy$

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3. Solve the Differential Equations

(a) Solve the equation:

$x^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} - 2y = 2 \ln(x)$

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(b) Find the complete integral using Charpit’s method for:

$0 = z(p + q) + b(p^2 + q^2 - 1)$

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4. Partial Derivatives

(a) Find all first-order partial derivatives of the function:

$h(x, y, t) = e^{-tx} + \cos(t)$

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5. Differential Equations

(a) Solve:

$\frac{d^2y}{dx^2} + 2x \frac{dy}{dx} - 16y = 0$

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6. Space Curve

Find the differential equations of the space curve where two surfaces intersect.

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7. Envelope and Characteristics

Find the envelope and characteristic curves of the family of curves.

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8. Method of Variation of Parameters

Solve the differential equation using the method of variation of parameters:

$\frac{d^2y}{dx^2} + 2y = \sec(x)$

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9. Limits of Functions

Prove if the limit of a given function exists at the origin.

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If you'd like detailed solutions for each of the above problems, let me know. Alternatively, I can focus on specific questions or steps you find challenging. Would you like detailed explanations for all the problems or a particular one?

Here are some follow-up questions you can consider:

1. How is Charpit’s method applied to solve partial differential equations?
2. Can you explain the variation of parameters method in differential equations?
3. What are the differences between linear and non-linear differential equations?
4. How do you compute first-order partial derivatives for multivariable functions?
5. What are the conditions for a differential equation to be integrable?

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Tip: For second-order differential equations, always check if a simpler substitution can reduce the complexity of the equation before solving.