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
  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.

Linear Equations With 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.).

Linear Equations With Constant Coefficients

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

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