and the solution of the di erential equation is sinx+ xlny+ ey = C 8. #57 Show that v= lnytransforms dy dx + P(x)y= Q(x)(ylny) into dv dx + P(x) = Q(x)v(x) Proof: Let v= lny! dv dx = 1 y dy dx! dy dx = y dv dx Then, the di erential equation dy dx + P(x)y= Q(x)(ylny) becomes y dv dx + P(x)y= Q(x)(yv(x)) Dividing both sides by yyields dv dx + P(x) = Q(x)v(x) 5

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Speed Time & Distance shortcut Tricks Pdf, “Speed Time & Distance Questions PDF” In this post we are providing you the Speed Time & Distance pdf with detailed solution & Short Linear Partial Di erential Equations 9 where the functions ˚and Sare real. Find the partial di erential equations are ˚and S. Solution 9. Since @ @t = and @2 @x2 j = we obtain the coupled system of partial di erential equations @ @t ˚2 + r(˚2rS)=0 @ @t rS+ (rSr)rS= 1 m r (~2=2m)r2˚ ˚ + rV : This is the Madelung representation of the Schr odinger equation.

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Problem 7. De ne the function y = cos(logx). Find positive constants a, b, and csuch that ax2 y00+ bxy0+ cy= 0. Solution. The derivative of y= coslogxis y0= sinlogx 1 x Solution.

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Applying the above formula for the general solution yields v(t) = 49 + ce t=5; notice it is not accurate to speak of the solution of this differential equation. This is because it has many solutions, namely ( )= 3 3+ where is any constant (the so-called “constant of integration”). It is not always as easy to find formulas for solutions of a di fferential equation as it is for the equation (0.1). principles; Green’s functions.