Solving dy/dx = y²cos(x)

Calculus 2 · Axiom Academy

Solve a separable first-order differential equation with an initial condition, step by step. Solve the differential equation below for y(x) , using the given initial condition to pin down the constant of integration. Nice work — you solved a separable differential equation with an initial condition from start to finish. Here is what carried the solution: Separation of variables: Gather every y (including dy ) on one side and every x (including dx ) on the other, turning into . Integrate each side: Use the power rule and — and never drop the constant of integration C . Solve for y : Clear the negative sign and take reciprocals to reach the general solution . Pin down C : Substitute . Since , the condition forces C = -1 . Particular solution: — verify it by checking y(0) = 1 and that it satisfies the original equation. Separation of variables is the workhorse method for first-order equations: separate, integrate, solve for y , then use the initial condition. Practice it on other separable equations to build speed.

This is the written version of the interactive lesson above. See the full Calculus 2 course.