# Linear Equations With Constant Coefficients

__अचर गुणांकों वाले रैखिक समीकरण (Linear Equations With Constant Coefficients):__

__अचर गुणांकों वाले रैखिक समीकरण (Linear Equations With Constant Coefficients):__- इस आर्टिकल में अचर गुणांकों वाले रैखिक समीकरण (Linear Equations With Constant Coefficients) के बारे में बताया गया है.Definition:A linear differential equation is an equation in which the dependent variable y

and its differential coefficients occur only in the first degree. The general

form of such an equation is __आपको यह जानकारी रोचक व ज्ञानवर्धक लगे तो अपने मित्रों के साथ इस गणित के आर्टिकल को शेयर करें ।यदि आप इस वेबसाइट पर पहली बार आए हैं तो वेबसाइट को फॉलो करें और ईमेल सब्सक्रिप्शन को भी फॉलो करें जिससे नए आर्टिकल का नोटिफिकेशन आपको मिल सके।यदि आर्टिकल पसन्द आए तो अपने मित्रों के साथ शेयर और लाईक करें जिससे वे भी लाभ उठाए।आपकी कोई समस्या हो या कोई सुझाव देना चाहते हैं तो कमेंट करके बताएं।इस आर्टिकल को पूरा पढ़ें।__

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- \left(\frac{d^{n}y}{dx^{n}}\right)+P_{1}\left(\frac{d^{n-1}y}{dx^{n-1}}\right)+P_{2}\left(\frac{d^{n-2}y}{dx^{n-2}}\right)+……..+P_{n}y=0 ……(i)

Where Q

and P_{1},P_{2},P_{3},…..,P_{n}are all

constants or functions of x.

If P_{1},P_{2},P_{3},…..,P_{n}

are all constants (Q may not be constant), then the equation is said to be a linear differential equation

constant coefficients.

- We shall

first of all consider the differential equation in which the second member viz

Q is zero - i.e. \left(\frac{d^{n}y}{dx^{n}}\right)+P_{1}\left(\frac{d^{n-1}y}{dx^{n-1}}\right)+P_{2}\left(\frac{d^{n-2}y}{dx^{n-2}}\right)+……..+P_{n}y=0 ……(ii)

\text{ If } y=f_1\left(x\right)

be a solution of (ii),then by substitution in (ii) it can be seen that y=cf_{1}\left(x\right),where

C is an arbitrary constant,is also a solution of (ii).

Similarly

if y=f_{2}\left(x\right),y=f_{3}\left(x\right)……..,y=f_{n}\left(x\right) are the

solutions of (ii),then y=C_{2}f_{2}\left(x\right),y=C_{3}f_{3}\left(x\right),….,y=C_{n}f_{n}\left(x\right),

where C_{2},C_{3},…..,C_{n} are arbitrary constants,are

also the solutions of (ii) Also substitution will show that

y=C_{1}f_{1}\left(x\right)+C_{2}f_{2}\left(x\right)+…+C_{n}f_{n}\left(x\right) ………(iii)

Is also a

solution of (ii) \text{ If } f_{1}\left(x\right),f_{2}\left(x\right),f_{3}\left(x\right),….are

linearly independent,then (iii) is the complete integral of (ii),since it

contains n arbitrary constants and (ii) is order n.

Now let

us consider the equation (i),in which the second member viz. Q is also zero.If

y=f(x)

be solution of (i),then

y=F(x)+f(x) be a solution of (i),then

where F(x)= C_{1}f_{1}\left(x\right)+C_{2}f_{2}\left(x\right)+…+C_{n}f_{n}\left(x\right)is also a solution of (i) since the

substitution of F(x) for y in the left hand member of (i) gives:zero and that of:f(x)

for y gives Q,as y=f(x)

is solution of (i).

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- solution (iv) contains n arbitrary constants and (i) is differential equation

of nth order,therefore it is the complete solution of (i). The part F(x) is called the complementary function (C.F.) and the part f(x) is called the (P.I.).

Also Read This Article:Linear differential equation

- उपर्युक्त आर्टिकल में अचर गुणांकों वाले रैखिक समीकरण (Linear Equations With Constant Coefficients) के बारे में बताया गया है.

#### Linear Equations With Constant Coefficients

# अचर गुणांकों वाले रैखिक समीकरण

(Linear Equations With Constant Coefficients)

### Linear Equations With Constant Coefficients

इस आर्टिकल में अचर गुणांकों वाले रैखिक समीकरण (Linear Equations With Constant Coefficients) के बारे में बताया गया है.

Definition:A linear differential equation is an equation in which the dependent variable y

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