Differential Equations

Step 1: Group the terms in pairs:
Step 2: Evaluate each pair:
Step 3: Add all negative terms:
Step 4: Add the remaining term:
Step 5: Find the total sum:

Step 1: Rewrite the given differential equation. Rearranging,
Step 2: Check if the equation is homogeneous. Let
Step 3: Substitute into the equation.
Step 4: Integrate both sides.
Step 5: Substitute back .
Step 6: Use the given point .
Step 7: Write the final equation.
Hence,

Step 1: Rewrite the integrand using powers. Step 2: Apply substitution. Let So the integral becomes: Step 3: Integrate using the power rule. Here, : Step 4: Substitute back. Step 5: Final conclusion.