Author name: Ankur Negi

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Q1: Find the general solution of the differential equation: Solution:Step 1: Simplify the Trigonometric Expression Recall the identities: Substitute these into the differential equation: … (1) Step 2: Separate the Variables Move the term to the right side to separate the variables : Step 3: Integrate Both Sides Step 4: Perform the Integration So, the

Q1: Verify that the given function is a solution of the differential equation . Solution:Step 1: Write down the given function Given function: … (1) Step 2: Find the First Derivative () Differentiating equation (1) with respect to : … (2) Step 3: Find the Second Derivative ()We differentiate equation (2) with respect to :

Q 1: Determine the order and degree (if defined) of the following differential equation: Solution:Step 1: Identify the Highest Order DerivativeLooking at the equation, we have a fourth-order derivative term: .We also have a lower order derivative inside the sine function.The highest order derivative is 4. Therefore, the Order is 4. Step 2: Check the

Q1: Find the area under the given curves and given lines: (i) , , and x-axis (ii) , , and x-axis The formula is straightforward: . Since Solution Both curves and are standard parabolas opening upwards and lie above the x-axis for positive , we don’t need to worry about absolute values or negative areas