GATE 2022 MA – Question 19

Ordinary Differential Equations · First Order Equations and Initial Value Problems · 1 mark · Multiple choice

For real y₀, the initial value problem y′=cos(xy),x∈ℝ,y(0)=y₀ has how many solutions?

A unique solution
Exactly two
Infinitely many
None

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Show answer and explanation

Correct answer: (A) A unique solution

Explanation

The right side is smooth in x,y, hence locally Lipschitz in y and locally unique. Its absolute value is at most 1, so |y(x)−y₀|≤|x| on finite intervals. The solution cannot blow up at a finite endpoint and extends to all ℝ. **A**.

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