# Solution for a second order differential equation

• anooja559
In summary, Anooja is seeking help with solving a second-order differential equation and has made some attempts using integration. Willem suggests approaching the problem by letting y=dM/dr and solving a first-order differential equation.
anooja559
Member has been advised to use the template and show some effort.
Hi,
∂M/r∂r+∂2M/∂r2 = A

[Moderator's note: Moved from a technical forum and thus no template.]

Last edited by a moderator:
anooja559 said:
Hi,
∂M/r∂r+∂2M/∂r2 = A
Hi Anooja, what have you done so far? Where are you stuck?

WWGD said:
Hi Anooja, what have you done so far? Where are you stuck?
Hi WWGD,
Thank you so much.
So I integrated the equation
∫(∂M/r∂r+∂2M/∂r2 )= ∫A ----(1)
ie ∫∂M/r∂r+∫∂2M/∂r2 = ∫A
1/r*M-∫-1/r2*M + ∂M/∂r = A*r+C1 (integral by parts)
2*M/r+∂M/∂r = A*r+C1 ----(2)
Integrating eq(2) again

∫2*M/r+∂M/∂r = ∫A*r+C1

Which gives,
2*M*ln(r)+M = A*r2+C1*r+C2

However, the reference shows the solution as M = Ar2+C1*ln(r)+C2

anooja559 said:
Integrating eq(2) again

∫2*M/r+∂M/∂r = ∫A*r+C1

Which gives,
2*M*ln(r)+M = A*r2+C1*r+C2
∫2*M/r isn't equal to 2M∫ (1/r).
I can't really see how to do it, except by just guessing $M = a r^b$

The reference solution isn't completely correct, it should be $$M = \frac{ Ar^2}{4} +C_1 ln(r) +C_2$$

willem2 said:
∫2*M/r isn't equal to 2M∫ (1/r).
I can't really see how to do it, except by just guessing $M = a r^b$

The reference solution isn't completely correct, it should be $$M = \frac{ Ar^2}{4} +C_1 ln(r) +C_2$$
Dear Willem,
That is great, Thanks a lot, Could you please elaborate on how to get this

If you let ##y = \frac{dM}{dr}##, you have the first-order differential equation
$$y' + \frac 1r y = A.$$ Can you solve that one?

## 1. What is a second order differential equation?

A second order differential equation is a mathematical equation that involves an independent variable, a dependent variable, and the derivatives of the dependent variable with respect to the independent variable. It is called "second order" because it contains the second derivative of the dependent variable.

## 2. What is the general form of a second order differential equation?

The general form of a second order differential equation is:
y'' + p(x)y' + q(x)y = r(x),
where y is the dependent variable, x is the independent variable, p(x), q(x) and r(x) are functions of x, and y' and y'' represent the first and second derivatives of y with respect to x, respectively.

## 3. How do you solve a second order differential equation?

There are several methods for solving a second order differential equation, including the method of undetermined coefficients, variation of parameters, and the Laplace transform method. The specific method used will depend on the form of the equation and the boundary conditions given.

## 4. What is the solution to a second order differential equation?

The solution to a second order differential equation is a function that satisfies the equation and the given boundary conditions. It is typically expressed in terms of the independent variable, x, and may involve arbitrary constants that are determined by the boundary conditions.

## 5. What are some real-life applications of second order differential equations?

Second order differential equations are used to model many physical phenomena, such as the motion of a pendulum, the growth of a population, and the behavior of electrical circuits. They are also used in engineering to analyze and design systems, such as control systems and vibration analysis.

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