What is the particular solution to the differential equation dy/dx =3 cos(x)y with the initial condition y(pi/2)=-2?

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What is the particular solution to the differential equation dy/dx =3 cos(x)y with the initial condition y(pi/2)=-2?

What is the particular solution to the differential equation dy/dx =3 cos(x)y with the initial condition y(pi/2)=-2?

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What is the particular solution to the differential equation dy/dx =3 cos(x)y with the initial condition y(pi/2)=-2?

Answer 1

Weseparate variables and integrate: ∫dy/y = ∫3cosxdx; lny = 3sinx + lnC; y = Ce^3sinx;

y(π/2) = Ce^3sinπ/2 = -2; C = -2e^-3;

So, y = -2e^{3sinx - 3}

Answer 2

A first order homogeneous linear differential equation is one of the form y˙+p(x)y=0 or equivalently y˙=−p(x)y

So Sarah, you have what I term a first order linear homogeneous diff. eq. - which has a standard procedure for obtaining homogeneous solution (the no input solution with some initial condition). I suppose you could either say there is no particular solution, or perhaps it is zero.

the homogeneous solution= constant *e to the power of the intergal of the coefficient of y (using sign same as when y is on opposite side of equation than y') inthis case y(x)=k*e^3sinx

let's check, y'=(3cosx)*k*e^3sinx

for k,,,,,, -2=k*e³ ,,,,k=-2/e³

if y=(-2/e³)e^3sinx, when x=pi/2, y=-2