Question

3. We consider differential equations of the form (a) Suppose we have found (by some means) a solution yi(t) of equation (1). Verify that y(t)-yi(t)+w(t) is also a solution of (1) where w is any function which satisfies Notice this shows that the general solution of (1) can be found if we know at least one particular solution of (1) and the general solution of (2). Moreover, the general solution of (2) can be found by using the method of the previous problem. (b) Find the general solution of given that one solution is y-t.

Answer 1

Given that The Differential Equation of the form y prime = p(t) + q(t) y + r(t) y^2 Now, y prime = p(t) + q(t) + r(t) y^2 If y_1(x) be the solution therefore y prime = p(t) + q(t) y + r(t) y^2 (1) Let y(t) = y_1(t) + w(t) where w(t) is a solution of w prime - (2 + 2r yi) w = rw^2 (2) Consider, y(t) = y_1(t) + w(t) y prime + y prime_1 + w prime y prime - p(t) - 2(t) y - r(t) y^2 y prime_1 + w prime - p(t) - q(t) [y_1 + w] - r(t) (y_1 + w]^2 y prime_1 + w prime - p(t) - q(t) y_1 - 2(t) w - r(t) [y^2_1 + 2wy_1 + w^2 y prime_1 + w prime - p(t) - q(t) y_1 - 2tw - r(t) [y^2_1 + 2wy_1 + w^2] y prime + w prime - p(t) - 2(t) y_1 - 2(t) w - r(t) w - r(t) y^2_1 - 2wry_1 - rw^2 [y^2_1 - p(t) - q(t) y_1 r(t) y^2_1]