Answer to Question #327262 in Differential Equations for n hasan

Question #327262

Find the differential equation whose solution is 𝑦 = 𝐴π‘₯ + 𝐡π‘₯2.


1
Expert's answer
2022-04-12T10:05:45-0400

Let's find yβ€²y' and yβ€²β€²y'' :

y=Ax+Bx2y=Ax+Bx^2

yβ€²=A+2Bxy'=A+2Bx (1)

yβ€²β€²=2By''=2B (2)

From (2) we have:

B=yβ€²β€²2B=\frac{y''}{2}

Putting B into (1) we obtain:

yβ€²=A+2yβ€²β€²2x=A+yβ€²β€²xy'=A+2\frac{y''}{2}x=A+y''x

A=yβ€²βˆ’yβ€²β€²xA=y'-y''x

Substituting A and B into the origin equation we get:

y=(yβ€²βˆ’yβ€²β€²x)x+yβ€²β€²2x2y=(y'-y''x)x+\frac{y''}{2}x^2

x22yβ€²β€²βˆ’xyβ€²+y=0\frac{x^2}{2}y''-xy'+y=0

Answer: x22yβ€²β€²βˆ’xyβ€²+y=0\frac{x^2}{2}y''-xy'+y=0 .


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