Differential Equations
Solving Differential Equations with Initial Conditions
Grade 12
Question:
<p>The equation of the curve passing through <span>\((3, 4)\)</span> and satisfying the differential equation <span>\(y\frac{dy}{dx} + (x^2 - y)\frac{dy}{dx} - x^2 = 0\)</span> can be</p>
<p>(A) <span>\(x^2 + y + 1 = 0\)</span></p>
<p>(B) <span>\(x^2 + y^2 = 25\)</span></p>
<p>(C) <span>\(x^2 + y^2 - 5x - 10 = 0\)</span></p>
<p>(D) <span>\(x + y - 7 = 0\)</span></p>
Step-by-Step Solution
Key Concept: Solve the differential equation and use the initial condition to find the particular solution, then verify the curve passes through the given point.
<p>Solve the given differential equation and apply the initial condition that the curve passes through <span>$(3, 4)$</span> to determine which option satisfies both conditions. Verify: <span>$3^2 + 4^2 = 9 + 16 = 25$</span>.</p>
Correct Answer: B