10:55 PM ODE OF 1ST ORDER AND 1ST DEGREE F(X,Y,Y') |
| الصورة العامة - y' = F(x,y) or M(x,y)dx + N(x,y)dy = 0 1- Separable equations : ويتم فيها فصل x عن y (فصل المتغيرات) to solve the separable equation : 1. set y' = dy/dx dy/dx = f(x)g(y) 2. dy/g(y) = f(x)dx 3. by integration we obtain the solution . :: Special case :: - If the DE in the form y' = F ( ax + by + c ) to solve this equation we set [ ax + by + c = z ] 2- Homogeneous equations : F(x,y) is called homogeneous of degree if F( tx, ty ) = tn F(x,y) إذا كان كل حد من حدود المعادلة من نفس الدرجة تبقى homogeneous or x/y or y/x a DE M(x,y)dx + N(x,y)dy = 0 is called homogeneous if M(x,y) & N(x,y) are homogeneous of the same degree To solve the homogeneous DE :- 1. set y = vx 2. y' = v + xv' 3. بالتعويض هاتتحول المعادلة الى separable :: Differential Equations reducible either separable or homogeneous :: if the differential equations in the form a1x + b1y + c1 y' = ـــــــــــــــــــــــــــــــــــــــ a2x + b2y + c2 1- parallel : if a1/b1 = a2/b2 we set a1x + b1y = z or a2x + b2y = z then the equation is reduced to separable . 2- intersection : if a1/b1 != a2/b2 we solve the two equation to find the point of intersection (h,k) then we set x = t + h , y = s + k and dy/dx = ds/dt by substitution , the equation is reduced to homogeneous . :: Exact differential equation :: A DE M(x,y)dx + N(x,y)dy = 0 لازم موجب if dM/dy = dN/dx the equation is called exact . the solution of the exact DE is in the form  حيث xo , yo قيم ابتدائية اختيارية يتم اختيارها على اساس جعل المعادلة أسهل ما يمكن ولا تعتمد على بعضها وعادة ما نختارها بـ 0 أو 1 في حالة ln :: Integrating Factor عامل المكاملة :: if dM/dy != dN/dx the DE is not exact . - we can transform this equation to exact by multiplying it by integrating factor M ميو - to find the integrating factor - two cases :-   :: Linear differential equations ::  :: Bernoulli 's equation ::  ___ , ___ , ___ , ___
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