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How to Solve Linear Differential Equation?

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Last updated date: 02nd Aug 2024
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Know About How to Solve Linear Differential Equation?

Linear differentiation equation is a form of the equation that is linear and polynomial. It is a part of the new derivatives and functions of an equation of a structure. A linear equation usually consists of derivatives of multiple variables. When a result and variable is partial, then it is also stated as a linear partial differential equation.

In an equation when its new function and the derivative is not in the linear form, it is then considered to be a nonlinear differential equation. It is used to find a solution for multiple chaoses.

In an equation where x dy / dx + Y = 0, it involves variables and a derivative which is dependent on y variable linked to independent variable x. This form of an equation is called a differential equation. Usually, an equation with a result of dependent variable connected to independent variables is known as a differential equation. It involves dependent variable derivatives concerning a single independent variable, also known as an ordinary differential equation.

Apart from these, students get to learn about the first-order nonlinear differential equation and first-order linear differential equation. Here the variables like P and Q works as an independent variable or remain constant.

The equation dy/dx + Py = Q can be a form of differential equation in a first-order linear differential. Here P and Q can be either function of y, which is an independent variable or constants only. A student has to derive the standard form or a depiction of this solution.

How to Solve First Order Nonlinear Differential Equation

A first-order differential equation is considered to be separable if it can be written in the form ϕ(y)dy = ψ(x)dx. This suggests the official process of anti differentiating ϕ(y) and ψ(x) concerning y and x. Moreover, results to arrive at an implicit general solution is equated as Z ϕ(y)dy = Z ψ(x)dx + C. It is important to note that here R ϕ(y)dy and R ψ(x)dx stand for definite antiderivatives of ϕ(y) and ψ(x).

The nonlinear differential equations definition can be understood from this explanation. In a nonlinear system of equations cannot be written or solved as a linear combination of new function or variable. A system can be termed as nonlinear despite a linear function shows in an equation. In other words, a differential equation can be linear when its the unknown function and derivatives are linear in terms. This will be same even if the other variables in terms of nonlinear appears.

Apart from the nonlinear differential equation, students need to practice equation based on linear equations. They should follow quality test papers and exercise materials, offering multiple questions on how to solve a nonlinear differential equation.

One can check Vedantu, which is one of the most prominent education portal offering multiple benefits. They provide solutions and an example of the nonlinear differential equation for revision purposes. Moreover, students can check their live classes and training sessions available for a budget-friendly price. To make the best of these features, download the official app today!

FAQs on How to Solve Linear Differential Equation?

1. What is the Primary Difference Between Linear and Nonlinear Equation?

Ans. A linear equation can be defined as a form of the equation that is linear polynomial in nature. It is a part of the new derivatives and functions of an equation of a structure. A linear equation usually consists of derivatives of multiple variables. When a result and variable is partial, then it is also stated as a linear partial differential equation. It forms a straight line on the graph, and its general form is x = ny + c. Here x and y are variables while n is a slope and c is the constant value.

While when a new function and the derivative is not in the linear form or not polynomial, it is then considered to be a nonlinear equation. It forms a curve on the graph as its general form is ax² + by² = c. Here a, b and ca are constant value and x, y is variable.

2. What is the Formula to Solve the First-Order Differential Equation?

Ans. A first-order differential equation is a sum in which ƒ(x, y) is a function of two variables. It is generally defined in an XY-plane region. The equation is of the first order because it occupies only the first derivative dy and dx.

The first-order differential equation is a form of a sum which can be shown as F(p, y', y ) = 0. The solution for this equation will be a function f(p) which makes F(p, f(p), f′(p)) = 0 for all the value of p. Moreover, F can be considered as a function of three variables which we can show as p, y' and y. It can be seen that ˙y  appears plainly in the equation though p and y are not required. First-order, this term here means the first derivative of y appears, but no higher order of results are seen.

3. What is the Answer for this Differential Equation: dy/dx + (sec x)y = 6?

Ans. Evaluating the sum where dy/dx + Py = Q. We consider,  P = sec  x, Q = 6

Here the integrating factor can be found by using the formula e∫Pdx= I.F and e∫secdx  = I.F.I.F. =  eln|secx + tanx|=secx + tanx

We can write LHS as d(y × I.F)/dx}, that is d(y × (sec x + tan x ). Now d(y × (sec x + tan x ))/dx =6(sec x + tan x).

Incorporating both the sides w, r, t,x,

We get ∫d(y × (secx + tanx)) = ∫6(secx + tanx)dx

y × (secx + tanx) = 6(ln|secx + tanx|+ log|secx|), y = 6(ln|secx + tanx|+ log|secx|(secx + tanx) + c.