The initial value problem dy/dx = x, y(0) = 1 hasa)only one solutionb)two solutionc)no solutiond)an infinite number of solutionsCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Here f(x,y) = x and (δf/δy) = 0 are both bounded in any neighborhood about the point (0,1) . So by Picard existence & uniqueness theorem we conclude that the solution of the given differential equation is unique.

Answer

Answer

Explanation:

Given Equation:
dy/dx = x, y(0) = 1

Existence and Uniqueness Theorem:
According to the Existence and Uniqueness Theorem for Ordinary Differential Equations, if the function x is continuous in some region around the initial condition, then there exists a unique solution to the initial value problem.

Analysis:
In this case, the function x = x is continuous everywhere, including around the point x = 0. Therefore, there exists a unique solution to the initial value problem dy/dx = x, y(0) = 1.

Conclusion:
Therefore, the initial value problem dy/dx = x, y(0) = 1 has only one solution.

Answer

Dy/dx=x
then. dy=xdx
integrating both side,we get
y=x^2/2+C
y(0)=1
then, 1=0+c
c=1
thus y=x^2/2+1
unique solution

Answer

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