Let y=y(x) be the solution curve of the differential equation (d y)/(d x)+((2 x²+11 x+13)/(x³+6 x²+11 x+6)) y=(x+3)/(x+1), x>-1, which passes through the point (0,1). Then y(1) is equal to :

Solution

Maths sir · 4 steps

Factor the cubic, split P(x) into partial fractions, and the integrating factor comes out as a neat fraction.

1. Factor and split. The denominator factors as (x+1)(x+2)(x+3). Putting x=-1,-2,-3 in turn gives (2x²+11x+13)/(x+1)(x+2)(x+3) = 2/(x+1) + 1/(x+2) - 1/(x+3).
2. Integrating factor. ∫ P dx = 2 ln(x+1) + ln(x+2) - ln(x+3), so the integrating factor is (x+1)²(x+2)/(x+3). For x>-1 every factor is positive, so no modulus signs are needed.

   y·(x+1)²(x+2)/(x+3) = ∫ (x+3)/(x+1)·(x+1)²(x+2)/(x+3) dx = ∫ (x+1)(x+2) dx = x³/3 + 3x²/2 + 2x + c

3. Find c. At (0,1) the integrating factor is (1 × 2)/3 = 2/3, so 1 × 2/3 = 0 + c, which gives c = 2/3.
4. So. At x=1 the integrating factor is (4 × 3)/4 = 3 and the right side is 1/3 + 3/2 + 2 + 2/3 = 9/2. So 3y(1) = 9/2, giving y(1) = 3/2, option (b).

The idea to keep. When P(x) is a rational function, factor the denominator and use partial fractions. The integrating factor then becomes a product of powers, and it often cancels most of Q(x).

Answer from the NTA final answer key. Solution by PrepWiser.