Question

Solution of Linear Differential Equation of First Order

Answer 1

Differential equation of first order

\(\frac{dy}{dx}\) + Py = Q ......(1)

Multiplying both side by f(x) function of x, we get

f(x) \(\frac{dy}{dx}\) + f(x)Py = f(x)Q .....(2)

Here, choosing function/in such a way that R.H.S. of equation (2), will be differential of yf(x).

Hence, multiplying equation (1) by f(x) = e^∫Pdx, L.H.S. will be differential coefficient of any function of x and y.

f(x) = e^∫Pdx is known as integrating factor (I.F.) of given differential equation.

Putting f(x) = e^∫Pdx in equation (2),

e^∫Pdx \(\frac{dy}{dx}\) + Pe^∫Pdxy = Qe^∫Pdx

or \(\frac{d}{dx}\) [e^∫Pdxy] = Qe^∫Pdx

Integrating both sides w.r.t. 'x', we get

\(\int\frac{dy}{dx}\) [e^∫Pdxy]dx = ∫Qe^∫Pdxdx

or ye^∫Pdx = ∫Qe^∫Pdxdx + C

or y = e^-∫Pdx[∫Q(e^∫Pdx)dx + C] ........(4)

Equation (4) is the general solution of differential equation (1).

Step for solving linear differential equation of first order:

• Step I. Write the given differential equation in the form \(\frac{dy}{dx}\) + Py = Q, where P and Q are constants or functions of x.
• Step II. Find the integrating factor (I.F.) = e^∫Pdx
• Step III. The differential equation of step I is multiplied by e^∫Pdx on both sides.
• Step IV. Now, we integrate the differential equation w.r.t. 'x'.
  ye^∫Pdx = ∫Qe^∫Pdx + C

         or y × (I.F.) = ∫Q (I.F.) dx + C

         which is the solution of differential equation.

Similarly, first order differential equation \(\frac{dy}{dx}\) + P₁y = Q₁, where PT and Q1 are constants or functions of y only, then for this

I.F. = e^∫P₁dx

and solution of the equation is

x(I.F.) = ∫(Q₁ x I.F.) dy + C

→ An equation involving derivatives of the independent variable with respect to independent variable (variables) is called 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.

→ Degree of a differential equation is the highest power (positive integer only) of differential coefficient.

→ In the differential equation of first order and first degree independent variable (x), dependent variable (y) and differential coefficient \((\frac{dy}{dx})\) of first order is involved. This type of equation is defined in the following form :

\(\frac{dy}{dx}\) = F(x, y) ........(1)

or \(\frac{dy}{dx} = \frac{h(x,y)}{g(x,y)}\) ......(ii)

where h(x, y) and g(x, y) is the function of x and y.
Simply, the differential equation of first order and first degree are written as :

P(x, y, \(\frac{dy}{dx}\)) = 0

→ Differential equation of first order and first degree of separable variable

\(\frac{dy}{dx}\) = h(x) × g(x)

or solution of \(\frac{dy}{g(y)}\) = h(x) dx

\(\int\frac{1}{g(y)}\) dy = ∫h(x)dx or G(y) = H(x) + C

where G(y) and H(x) are antiderivatives of \(\frac{1}{g(y)}\) h(x).

→ If differential equation of first order and first degree is defined as:

\(\frac{dy}{dx} = \frac{f(x,y)}{g(x,y)}\)....(i)

where f(x, y) and g(x, y) is the homogeneous equation of same degree then differential equation (1) is called as homogeneous differential equation.

For finding its solution, it is shown in the following way:

im-9

On substituting v = \(\frac{y}{x}\) in equation (iii) the general solution of the differential equation is obtained.

→ A differential equation of the form

\(\frac{dy}{dx}\) + Py = Q

where P and Q are constants or functions of x only is called a first order linear differential equation.

→ Step involved to solve first order linear differential equation:

• Step I. Write the given differential equation in the form
• Step II. Find the integrating factor (IF) = e^∫Pdx
• Step III. Multiply by e^∫Pdx in both sides of differential equation obtained by step I.
• Step IV. Write the solution of the given differential equation as
  ye∫PdX = ∫Qe^∫Pdx + C
  or y × (IF) = ∫Q (IF) dx + C

Similarly, the first order linear differential equation is in

\(\frac{dy}{dx}\) + P₁y = Q₁

where P₁ and Q₁ are constants or functions of y only. Then

I.F. = e∫P₁dy

and the solution of the differential equation is given by

x(I.F.) = ∫Q₁ × (I.F)dy + C