3.8.4—Differential equations

A solution of an ODE is a function whose derivatives satisfy the equation on the stated interval. Solve using the appropriate method, then apply initial or boundary information.

Check differentiability and any division restrictions. Substitution into the original equation is the strongest final check, especially after squaring or taking logarithms.

For y′=y, y=Ce^x satisfies the equation; y(0)=2 selects C=2.

A formula that matches one initial point is not enough; it must also satisfy the differential relation throughout the domain.