Can anyone help me solve this equation using the method differentiation with integrating factor:
dy/dt=(y-1)t^2, where inital condition is y(1)=2
I have to solve this equation using the method of integrating factor.
And i have no idea how
i know i have to change the equation to the form to linear
A1(t)dy/dt + A0(t)y=f(t)
then write it as: dy/dt + p(t)y=q(t)
where p(t)=A0(t)/A1(t) and q(t)=f(t)/A1(t)
Then solve by finding an integrating factor etc
Write the equation as . An "integrating factor" is a function u(t) such that mutiplying the entire equation by it:
will make the left side an "exact" derivative: .
By the product rule, and we want that equal to . That means we must have which is a separable equation for u: . Integrating both sides, (since we are only looking for a single solution we can ignore the constant of integration) or .
Multiplying the original equation by that gives .
Of course, the left side of that is a single derivative:
Integrate both sides of that (The left side is easy. Use the substitution on the right.) and solve for y.