How do we solve d^2/dx^2 + y = sinx*sin 2x
How do we simplify the RHS to the form sin(ax+b) ?
I do not understand your leftside
but sin 2x = 2 sin x cos x
so
sin x sin 2x = 2 sin^2 x cos x
so you have
d^2/dx^2 + y = 2 sin^2 x cos x
well d/dx ( sin^3 x )= 3 sin^2x cos x
so (2/3) d/dx (sin^3 x) =2 sin^2 x cos x (oh my)
d^2/dx^2 + y = (2/3) d/dx (sin^3 x)
maybe that might help
That's d^2y/dx^2 + y = sinx*sin 2x
Could you please explain how to get the particular integral of that?
There is no way to simplify sinx sin2x to sin(ax+b)
just a glance at the graph will tell you that much. It is not a simple sinusoidal curve.
To solve the differential equation d^2/dx^2 + y = sin(x) * sin(2x), we first need to simplify the right-hand side (RHS) to the form sin(ax + b).
Let's break down the process step by step:
Step 1: Expand the product of sin(x) * sin(2x) using the double angle formula:
sin(x) * sin(2x) = 1/2 * (cos(x - 2x) - cos(x + 2x))
= 1/2 * (cos(-x) - cos(3x))
= 1/2 * (cos(x) - cos(3x))
Step 2: Rearrange the terms to rewrite the RHS in the form sin(ax + b):
sin(x) * sin(2x) = 1/2 * (cos(x) - cos(3x))
= 1/2 * (-cos(-x) + cos(3x))
= -1/2 * (cos(-x) - cos(3x))
Comparing this expression with the form sin(ax + b), we can see that a = 1 and b = -x.
Hence, the simplified RHS in the form sin(ax + b) is -1/2 * sin(x + (-x)).
Now that we have simplified the RHS, we can proceed to solve the differential equation d^2/dx^2 + y = -1/2 * sin(x + (-x)).