# Linear Differential Equation and Equations Reducible to Linear Form

5 de Feb de 2023
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### Linear Differential Equation and Equations Reducible to Linear Form

• 1. Central University Of Kashmir Tulmulla Campus, Ganderbal Kashmir BT – 301 Linear differential Equation and Equations reducible to linear form Presented by: Sameem Makhdoomi Enrollment no. 2124CUkmr09
• 2.  What Is a Linear Differential Equation.  Steps to Solve Linear Differential Equation.  Different forms of LDE.  Examples on LDE.  Equations reducible to the linear form.  Examples on Bernoulli’s equation. Contents
• 3. The linear differential equation is of the form dy/dx + Py = Q, where P and Q are numeric constants or functions in x. It consists of a y and a derivative of y. The differential is a first-order differentiation and is called the first-order linear differential equation. What Is a Linear Differential Equation?
• 4. Steps to Solve Linear Differential Equation The following three simple steps are helpful to write the general solutions of a linear differential equation. Step - I: Simplify and write the given differential equation in the form dy/dx + Py = Q where P and Q are numeric constants or functions in x. Step - II: Find the Integrating Factor of the linear differential equation IF = e^∫P.dx Step - III: Now we can write the solution of the linear differential equation as follows.
• 5. Two forms:
• 6. Example 1
• 8. Example 2
• 9. IF /
• 11. Example 3
• 15. This can be simplified to represent the following linear differential equation. dy/dx - y/x = 2x Comparing this with the differential equation (dy/dx + Py = Q) we have the values of P = -1/x and the value of Q = 2x. Hence we have the integration factor as IF = e^∫ (−1x)dx IF = e^(-logx) = e^(logx^-1)= 1/x Example 4 xdy -(y + 2x^2).dx = 0
• 16. Further, the solution of the differential equation is as follows. y.(1/x) = ∫ 2x.(1/x) dx + c y/x = ∫ 2.dx + c y/x = 2x + c y = 2x^2 + xc -Ans
• 17. Equations reducible to the linear form (Bernoulli’s equation) dx
• 18. Example 5
• 19. Example 6
• 20. Example 7
• 21. Thank you Sameem Makhdoomi