Sec 1.5: Linear First-Order DE

Sec 1.5: Linear First-Order DE

Sec 2.3 Sec 1.5: Linear First-Order DE Definition 2.2 A 1st order De of the form is said to be linear first-order equation. 1 2 3

Sec 1.5 How to Solve ? Method of Solution: Rewrite into standard form (coeff of y’ is 1). Step 1 ----- (1) Find the Integrating Factor = ( ignore constant of integration) Step 2 Step 3 Multiply (1) by ( check: ) Integrate both sides: Step 4 ( DONOT forget constant of integration) 2 3 1

Sec 2.3 Remark: linear in y or linear in x Solve the IVP Find a general solution of Find a general solution of

Derivation The solution of the above DE is given: where

Theorem 1: Theorem 1: The Linear First-Order Equation Remarks: If the function and are continuous on the open interval I containing the point , then the initial value problem Theorem 1 gives a solution on the entire interval I Theorem 1 tells us every solution is included in the formula (6) Theorem 1 tells us that the general solution is given in(6) Theorem 1 tells us that a linear first-order DE has no singular sol has a unique solution y(x) on I, given by the formula in Eq(6) with an appropriate value of C. (6) where

Sec 2.3 Solve the IVP 1

Derivative and Integration D[x^3+3 x,{x,2}] MATHEMATICA Integrate[ x^2 , x] syms x diff(x^3+3 x) syms x int(x^2,x) MATLAB

Sec 1.5: Linear First-Order DE