The solution of the differential equation x ~d ^2 y d x^2 =1 at x= y =1 with dy d x =0 at x=1 , is

Step-by-step solution

The differential equation x d ^2 y d x^2 = 1 is solved by isolating the second derivative: d ^2 y d x^2 = 1 x Integrating once with respect to x yields the first derivative: d y d x = 1 x d x = x + C₁ Applying the initial condition d y d x = 0 at x = 1 determines C₁ : 0 = 1 + C₁ = 0 + C₁ C₁ = 0 The first derivative simplifies to d y d x = x . Integrating again using integration by parts with u = x and d v = d x gives: y = x d x = x x - x + C₂ Applying the initial condition y = 1 at x = 1 determines C₂ : 1 = 1 1 - 1