# What is the general solution of the differential equation # x^2y'' -xy'-3y=0 #?

see Below

and

Substitute this values in the original equation:

And:

Although this is suprisingly long, this actually simplifies to:

Integrate both sides:

Undo substitution:

Integrate both sides again:

Which is correct.

By signing up, you agree to our Terms of Service and Privacy Policy

# y=A/x + Bx^3#

We have:

This is a Euler-Cauchy Equation which is typically solved via a change of variable. Consider the substitution:

Then we have,

Substituting into the initial DE [A] we get:

This is now a second order linear homogeneous Differentiation Equation. The standard approach is to look at the Auxiliary Equation, which is the quadratic equation with the coefficients of the derivatives, i.e.

We can solve this quadratic equation, and we get two real and distinct solutions:

Thus the Homogeneous equation [B]:

has the solution:

Now we initially used a change of variable:

So restoring this change of variable we get:

Which is the General Solution

By signing up, you agree to our Terms of Service and Privacy Policy

The general solution of the given differential equation (x^2y'' - xy' - 3y = 0) is (y = c_1x^3 + c_2x^{-1}), where (c_1) and (c_2) are arbitrary constants.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the instantaneous velocity for the particle whose position at time #t# is given by #s(t)=3t^2+5t# ?
- How do you find the volume bounded by #f(x) = x^2 + 1# and #g(x) = x + 3# revolved about the x-axis?
- How do you find the particular solution to #(du)/(dv)=uvsinv^2# that satisfies u(0)=1?
- What is the arc length of #f(x)=xlnx # in the interval #[1,e^2]#?
- How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y=8 sqrt x#, y=0, x=1 revolved about the x=-4?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7