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2. (a) Show that the substitution vln y transforms the differential equationQ(x) (y In y),yP(x)yinto the linear equationo'- Q(x)v = -P(x)(b) Use this substitution to solve the differential equationry' -4xy2y In y = 0.3. Consider the initial value nroblem

Question
2. (a) Show that the substitution v
ln y transforms the differential equation
Q(x) (y In y),
yP(x)y
into the linear equation
o'- Q(x)v = -P(x)
(b) Use this substitution to solve the differential equation
ry' -4xy2y In y = 0.
3. Consider the initial value nroblem
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2. (a) Show that the substitution v ln y transforms the differential equation Q(x) (y In y), yP(x)y into the linear equation o'- Q(x)v = -P(x) (b) Use this substitution to solve the differential equation ry' -4xy2y In y = 0. 3. Consider the initial value nroblem

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Step 1

To transform the given differential equation using the appropriate substitution, and hence solve the simpler equation

Step 2

(a) Effect of substitution on the derivatives of y and v.

(a)v =lny
yye(chainrule)
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(a)v =lny yye(chainrule)

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Step 3

(a) verifi...

So. y'P(x)yx)y ln y)
transforms to
e'v' P(x)e' Qx)e'v
(linear equation)
v-Q)vPx)
as required
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So. y'P(x)yx)y ln y) transforms to e'v' P(x)e' Qx)e'v (linear equation) v-Q)vPx) as required

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