If the solution curve, of the differential equation (d y)/(d x)=(x+y-2)/(x-y) passing through the point (2,1) is tan ⁻¹((y-1)/(x-1))-1/β log ₑ(α+((y-1)/(x-1))²)=log ₑ|x-1|, then 5 β+α is equal to __________.

Solution

Maths sir · 4 steps

Shift the origin to where the two lines meet, and the equation turns homogeneous.

1. Find the new origin. Solve x + y - 2 = 0 and x - y = 0 together to get x = y = 1. Put X = x - 1 and Y = y - 1. The equation becomes dY/dX = (X + Y)/(X - Y), which is homogeneous.
2. Substitute Y = vX. Then dY/dX = v + Xdv/dX, so Xdv/dX = (1+v)/(1-v) - v = (1 + v²)/(1 - v). The variables now separate.

   ∫ (1 - v)/(1 + v²) dv = ∫ dX/X ⇒ tan⁻¹ v - 1/2 ln(1 + v²) = ln|X| + C

3. Use the point. At (2, 1) we have X = 1 and Y = 0, so v = 0. That gives 0 - 0 = 0 + C, so C = 0. With v = (y-1)/(x-1) and X = x - 1, this matches the given curve exactly.
4. So. Comparing, α = 1 and β = 2. Then 5β + α = 10 + 1 = 11.

The common slip: 6. Integrating v/(1+v²) as ln(1+v²) and dropping the 1/2 makes β = 1, which gives 6.

The idea to keep. When the numerator and denominator are linear with constant terms, move the origin to the point where the two lines meet. Then Y = vX finishes the job.

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