# Differential equations of first order

Student à Universal College of Engineering and Technology
29 Aug 2015
1 sur 15

### Differential equations of first order

• 1. What is Differential equation?  If y is a function of x, then we denote it as y = f(x). Here x is called an independent variable and y is called a dependent variable.  If there is a equation dy/dx = g(x) ,then this equation contains the variable x and derivative of y w.r.t x. This type of an equation is known as a Differential Equation.
• 2. Order of Differential Equation  Order of the highest order derivative of the dependent variable with respect to the independent variable occurring in a given differential equation is called the order of differential equation.  E.g. – 1st order equation  2nd order equation
• 3. Degree of Differential Equation  When a differential equation is in a polynomial form in derivatives, the highest power of the highest order derivative occuring in the differential equation is called the degree of the differential equation.  E.g. – Degree – 1 ,(d²y/dx) + dy/dx = 0 Degree – 2 , (d²y/dx)² + dy/dx = 0
• 4. Solution of differential equations of the first order and first degree  Differential equations of 1st order can be solved by many methods ,some of the methods are as follows :- 1. Variable Separable Method 2. Exact equation method 3. Homogenous equation method 4. Linear equation method
• 5. Solution of differential equations of the first order and first degree 5. Non-Linear Equation method (Bernoulli's equation) 6. Non-Exact Equation method
• 6. ( , )y f x y 
• 7. Important Forms of the method  Here are some important forms of the method through which we can know the form of equation and then use or apply the method which is required :- 1. Variable Separable method – Equation is in the form of : dy/dx = M(x)/N(y) or dy/dx = M(x)N(y)
• 8. Important Forms of the method 2. Exact equation method – equation is in the form of : Mdx + Ndy = 0 --- 1 If , ∂M/∂y = ∂N/∂x Then the above equation 1 is Exact equation 3. Homogenous equation method - equation is in the form of : dy/dx = x²y + x³y + xy²/x³ - y³ (Example)
• 9. Important Forms of the method 4. Linear equation method - equation is in the form of : Form -1 : dy/dx + Py = Q (x form) Form - 2 : dx/dy + Px = Q (y form) 5. Non-Linear Equation method (Bernoulli's equation) - equation is in the form of : Dy/dx – 2ytanx = y²tan²x (Example)
• 10. Important Forms of the method 6. Non-Exact Equation method - equation is in the form of : Type – 1 : Mdx + Ndy = 0 F(x) = 1/N (∂M/∂y - ∂N/∂x) (x form) Type – 2 : Mdx + Ndy = 0 F(y) = 1/M (∂N/∂x - ∂M/∂y)
• 11. 1st Order DE - Homogeneous Equations Homogeneous Function f (x,y) is called homogenous of degree n if :    y,xfy,xf n   Examples:   yxxy,xf 34   homogeneous of degree 4            yxfyxx yxxyxf , , 4344 34       yxxyxf cossin, 2   non-homogeneous              yxf yxx yxxyxf n , cossin cossin, 22 2      
• 12. 1st Order DE - Homogeneous Equations The differential equation M(x,y)dx + N(x,y)dy = 0 is homogeneous if M(x,y) and N(x,y) are homogeneous and of the same degree Solution : 1. Use the transformation to : dvxdxvdyvxy  2. The equation become separable equation:     0,,  dvvxQdxvxP 3. Use solution method for separable equation         Cdv vg vg dx xf xf   1 2 2 1 4. After integrating, v is replaced by y/x
• 13. Variable – separable example 1. dy/dx = x.y =dy/y = xdx =∫dy/y = ∫xdx =logy = x²/2 + c
• 14. Exact Equation Example 1. xdy/dx + y + 1 = 0 =xdy + (y + 1)dx = 0 here , M = y + 1 , N = x ∂M/∂y = 1 , ∂N/∂x = 1 therefore , ∂M/∂y = ∂N/∂x here the given equation is an exact equation ∫Mdx(y constant) + ∫(terms of N not containing x)dy = c
• 15. Exact Equation Example =∫(y + 1)dx (y constant) + ∫0.dx = c = x(y + 1) = c