Let y=y₁(x) and y=y₂(x) be the solution curves of the differential equation (d y)/(d x)=y+7 with initial conditions y₁(0)=0 and y₂(0)=1 respectively. Then the curves y=y₁(x) and y=y₂(x) intersect at

Solution

Maths sir · 4 steps

Solve once with a constant C, plug in both starting values, then see if eˣ can ever balance them.

1. Separate the variables. Write the equation as dy/(y+7) = dx and integrate both sides. This gives ln|y+7| = x + c.

   y + 7 = Ceˣ

2. Use each starting value. For y₁(0)=0 you get C = 7, so y₁ = 7eˣ - 7. For y₂(0)=1 you get C = 8, so y₂ = 8eˣ - 7.
3. Try to meet. Setting y₁ = y₂ gives 7eˣ = 8eˣ, that is eˣ = 0. But eˣ > 0 for every real x, so this equation has no solution.
4. So. The two curves never intersect. The answer is (a), no point.

The idea to keep. For dy/dx = f(y) with a smooth f, two solution curves that start at different points can never cross. If they met, they would have to be the same curve.

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