Introduction to Differential Equations and Their Solutions

Differential Equations In this chapter, we will study some basic concepts related to differential equation, general and particular solutions of a differential equation, formation of differential equations, some methods to solve a first order first degree differential equation and some applications of differential equations in different areas. An equation involving derivative (derivatives) of the dependent variable (variables) with respect to independent variable (variables) is called a differential equation. A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation (ODE). Order and degree of a differential equation Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Degree is the power of the highest order derivative involved in the given differential equation. Consider the following differential equations: first order & first degree second order & first degree ( ) third order &first degree third order & its degree is not defined. Denote & and so on…… & 1 Try to solve General and Particular Solutions of a Differential Equation Example 1 Verify is a solution of the differential equation Solution since Substituting in DE ( L.H.S ) =R.H.S Therefore, the given function is a general solution of the given differential equation. 2 Example 2 Verify is a solution of the differential equation Solution since Substituting in DE ( L.H.S= ) =R.H.S Formation of a Differential Equation whose General Solution is given Suppose center at (–1, 2) and radius 1 unit. , represents a circle having Differentiating equation of circle with respect to x, we get , which is a differential equation (DE). Example 3 Form the differential equation representing the family of curves where, m is arbitrary constant. Solution , differentiating both sides of equation with respect to x, we get Substituting the value of m in equation, we get which is free from the parameter m and hence this is the required DE. 3 Example 4 Form the differential equation representing the family of curves where, a, b are arbitrary constants. Solution , Differentiating both sides of equation with respect to x, we get ( ) Eliminating a and b from equations Which is free from the arbitrary constants a and b and hence this required differential equation. Try to solve Form the differential equation representing the family of curves 12- A 4 Methods of Solving First Order, First Degree Differential Equations 1. Separation of Variables A first order-first degree differential equation is of the form ( ) (1) If ( ) can be expressed as a product ( ) ( ), where, ( ) is a function of and ( ) is a function of , then the differential equation (1) is said to be of variable separable type. The differential equation (1) then has the form ( ) ( ) If ( ) (2) , separating the variables, (2) can be rewritten as ( ) ( ) (3) Integrating both sides of (3), we get ∫ ∫ ( ) ( ) (4) Thus, (4) provides the solutions of given differential equation in the form ( ) ( ) Example 1 Find the general solution of the differential equation Solution Separating the variables ( ) ( Integrating both sides or which is the general solution of equation. 5 ) Example 2 Find the general solution of the differential equation Solution Separating the variables Integrating both sides ∫ ∫ which is the general solution of equation. Example 3 Find the general solution of the differential equation given that y = 1, when x = 0. Solution Separating the variables Integrating both sides ∫ ∫ or Substituting y = 1 and x = 0 in equation, we get we get the particular solution of the given differential equation as Example 4 Find the equation of the curve passing through the point (1, 1) whose ( ) differential equation is 6 Solution ( Separating the variables Integrating both sides ∫ ) ∫( ) ( | | ) at the point (1, 1) ( | |) Try to solve 12345- Choose the correct answer 7 2. Homogeneous Differential Equations The following function in and ( ( ) ) is a homogeneous function if ( ( A differential equation of the form if ( ) )is said to be homogenous )is a homogenous function of degree zero. To solve a homogeneous differential equation of the type ) …(1) ( …..(2) We make the substitution Differentiating equation (2) with respect to x, we get ….(3) Substituting the value of from equation (3) in equation (1), we get ….(4) ( ) ( ) or Separating the variables in equation (4), we get ( ( ) …..(5) ) Integrating both sides of equation (5), we get ∫( ( ) ∫ ) +C …(6) Equation (6) gives general solution (primitive) of the differential equation (1) when we replace by Example 5 Solve the following differential equation ( Solution Equation is homogeneous 8 ) ( ) ( ) ( ∫ ( ) ( ) ( ) ( ) ) ( ) separate variables ∫ | | | | | | | | put Example 6 Solve the following differential equation ( ) Solution Let ( ) ( ) ( ) ( ) ( ) ) is a homogenous function of degree zero. So, the given Therefore, ( differential equation is a homogenous differential equation. ( ) ( ) ( ) ( ) 9 ( ) ( ) ( ( ) Separate variables ) ( ) ∫ ( ) ( ∫ ) ∫( ∫( ) ) ( ) √( √( put ∫ ) ) Example 7 Solve the following differential equation ( Solution ( ) ( ) Equation is homogeneous Put ( ( ( ) ( ) ) ) ( ) Separate variables ∫ ∫ 11 ) (| |) ( ( put | | ) ) ( ( ) ) Try to solve 11 3. Exact Differential Equations The ordinary differential equation ( ) ( ) is Exact if The general solution is ∫ ( ) ∫ ( ) Example 8 Solve the following differential equation ( ) ( ) Solution ( ) and ( ) and the differential equation is exact ∫ ( ) ∫ ( ) ‫الحد المشترك بين التكاملين يكتب مرة واحدة‬ ‫فى الحل العام‬ The general solution is Example 9 Solve the following differential equation ( ( ) ) Solution ( ) and ( 12 ) and the differential equation is exact ∫ ( ) ∫ ( ) The general solution is 4. Reducible to Exact Differential Equations Let the ordinary differential equation is in the form ( If ) ( …(1) ) , then the differential equation is not exact. We calculate the integrating factor ( ) to transform differential equation to exact. Multiplication equation (1) by ( ( ) ( ), we get ) ) ( ( …(2) ) Then the equation (2) is exact. We have two cases for integrating factor: Case 1 [ ] Case 2 Example 10 Solve the following differential equation ( ) 13 (*) Solution ( ) ( and ) and the differential equation is not exact. [ ] ( ) Integrate both sides Multiplication equation (*) by ( ) ( ) ( ∫ ( ∫ ( ) ) ) ∫( ∫( ) ) The general solution is Example 11 Solve the following differential equation ( ) (*) Solution ( ) ( ) and ( ) and 14 the differential equation is not exact. [ ] ( ) Integrate both sides Multiplication equation (*) by ( ) ( ∫ ( ) ) ∫ ( ∫( ) ) ∫ The general solution is Try to solve 1. ( 2. ( ) ) ( 3. ) 4. ( ) 5. ( 6. ) ( ( 8. ( ) ) ) 7. ( ( ( ) ) ) 15 5. Linear Differential Equations Let the ordinary differential equation is in the form ( ) ( ) First, we determine ∫ ( ) ( ) , it is called Integrating factor The general solution is ( ) ∫ ( ) ( ) Example 12 Solve the following differential equation , with Solution ( ) ( ) ∫ ( ) ( ) and ∫ The general solution is ∫ ( 16 ) Example 13 Solve the following differential equation Solution ( ) ∫ ( ) ( ) and ∫ The general solution is ∫ We integrate by parts Example 14 Solve the following differential equation Solution ( ) and ( ) ( ) ∫ The general solution is ∫ 17 ( ) Try to solve 1. 2. ( ) ( ) ( ( ) ) 3. 4. 5. 18