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Differential equations of first order Definition of differential equation

- 1. Differential equations of first order
- 2. Introduction Definition of differential equation Classification of differential equations The order and the degree of a differential equation Classification of first order differential equation
- 3. Definition of differential equation An equation involving differentials or differential coefficients is called a differential equation. Thus, 1) dy = sin x dx 2) d y/dx = 0 3) y = x dy/dx + a/dy/dx 4) ∂z/∂x + ∂z/∂y = 1 5) ∂ z/∂x + ∂ z/∂y = 0 2 2 2 2 2 2
- 4. Classification of differential equations 1) Ordinary differential equations. 2) Partial differential equations. Ordinary differential equations : ordinary differential equations are those which involves ordinary derivatives with respect to a single independent variable. Thus equations, 1) dy = sin x dx 2) d y/dx = 0 3) y = x dy/dx + a/dy/dx 2 2
- 5. Partial differential equations : partial differential equations are those which involves partial derivatives with respects to two or more independent variables. Thus equations, 1) ∂z/∂x + ∂z/∂y = 1 2) ∂ z/∂x + ∂ z/∂y = 0 2 2 2 2
- 6. The order and the degree of a differential equation The order of the differential equation is the order of the highest derivative appearing in the differential equation. The degree of a differential equation is the degree of the highest derivative, when the derivatives are free from radicals and fractions. Example : ( d y/dx ) + (dy/dx) = c order : 2 degree : 2 2 22 3
- 7. Formation of a differential equation Ordinary differential equations are formed by elimination of arbitrary constants. Example : from the differential equation of simple harmonic motion given by, x = a sin (ωt + ) Solution : there are two arbitrary constants a and therefore, we differentiate it twice w.r.t. t, we have, dx/dt = ωa cos (ωt + ) and d x/dt = -ω a sin (ωt + ) = -ω x thus, d x/dt + ω x = 0 which is the required d.e. 2 2 2 2 2 2 2
- 8. Classification of first order differential equation 1) Variable separable. 2) Homogeneous equations. 3) Linear equations. 4) Exact equations.
- 9. Variable separable method : the general form of this type of equation is M(x) dx + N(y) dy = 0 Which can be solved by direct integration as ʃ M(x) dx + ʃ N(y) dy = c Example (1) : x dx + siny dy = 0 ʃ x dx + ʃ siny dy = 0 x /3 + ( -cosy ) = c (2) : 9y y + 4x = 0 9y dy/dx + 4x = 0 ʃ 9y dy + ʃ 4x dx = 0 9 y /2 + 4 x /2 = c 2 2 3 І 2 2
- 10. Homogeneous equations :An equation of the form dy/dx = f ( x, y ) / f ( x, y ) is called a homogeneous differential equation if f ( x, y) and f ( x, y ) are homogeneous functions of the same degree in x and y. Method of solution : 1) Put y = vx dy/dx = v + x dv/dx 2) Separate the variables in the new equation formed and solve. 1 1 2 2 . . .
- 11. Example : solve (x - y ) dy = 2xy dx Solution : dy/dx = 2xy/x - y put y = vx dy/dx = v + x dy/dx therefore v + x dv/dx = 2v/1-v or x dv/dx = 2v/1-v - v = v + v / 1 – v or 1 – v / v(1 + v ) dv = dx/x or (1/v – 2v/1 + v ) dv = dx/x integrating, we get log v – log (1 + v ) = log x + log c or log ( v /1 + v ) = log cx 2 2 2 2 2 2 . . . 3 2 22 2 2 2
- 12. or v/1 + v = cx or (y/x) 1/1 + (y/x) = cx or y/x x /x + y = cx or y = c ( x + y ) or x + y - 1/c y = 0 or x + y - by = 0 which is required solution. 2 2 . 2 2 2 2 2 2 2 2 2