# Download e-book for kindle: Ordinary Non-Linear Differential Equations by N.W. Mclachlan

By N.W. Mclachlan

ISBN-10: 1406742465

ISBN-13: 9781406742466

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4, so 6 integration. 2 dv where (8) two constants of are the a function of these constants. 221. ). , x() = (7) a/(l+bx) we find that 9 - tf_B+alog{xl(l+aAx)}]. (8) y is a function of A, B the constants of integration. 23. Solve = = 3 y"+3yy'+y =/(). 23 = Write u eSv dx then , = wy> = uv + uy' = 2 _= and Thus by (1), (3) we (3) , 3 = 4 0- HZ are linearly independent solutions of arbitrary constants, the complete solution is , =A we get y = u Since u = e I v dx , form in the where A =A , A u 1 +A 2 u 2 +A 3 u 3 1 =A 2 , 0, .

B. Assumed cubical-parabolic relation between anode current and grid potential. The working part of the curve lies between P and Q O is the centre of oscillation. e. the current tends to zero at P, and to a saturation value at Q. 9. A. Circuit ; round the grid circuit must vanish, so the differential equation in the absence of grid current is Idt-M I 0. dt (1) M The signof is minus, this being a known condition for self-oscillation. Assume that the anode current is given by Ia = <,[Eg -E*l*Et}, (2)t the transconductance of the valve, and Es is the grid potential corresponding to the anode saturation current.

14. Fig. 5 that the solution curve y of = coat. 10. (1), ~ cn ^ It is evident may from be analysed into 0039 a Fourier series. Its shape is such that in a first approximation we may write y where \A\ > |4 |. 10, (1), with = cot we get o) y ay = a(A In the 2 l (^ 1 cos last line, cos (2) terms involving A% (except two) have been omitted as negligible. Equating the coefficient of cos cu since Al ^ 0. +i64 ^ It follows 1 from (3) (3) 8, that the frequency is dependent EQUATIONS INTEGRABLE BY 30 upon the amplitude of the motion, which OH.