<\/span><\/h3>\n\nFind the solution to the differential equation y'(x) = x^2, with the initial condition y(0) = 3.<\/p>\n
To solve this problem, we first need to integrate the given differential equation:<\/p>\n
\u222bdy = \u222bx^2 dx<\/p>\n
This yields:<\/p>\n
y = (1\/3)x^3 + C<\/p>\n
where C is the constant of integration. <\/p>\n
Now we can use the initial condition y(0) = 3 to determine the value of C. Substituting the values into the equation, we have:<\/p>\n
3 = (1\/3)(0)^3 + C
\n3 = C<\/p>\n
Thus, the particular solution to the initial value problem is:<\/p>\n
y = (1\/3)x^3 + 3<\/b><\/p>\n