I want to find the solution of the following differential equation $ $ \frac{dy}{dx}+y^m=1.$ $ For example, if $ m=1$ , then $ y=1-e^{-x}.$ If $ m=2$ , we have $ y=\tanh(x)$ , but for $ m\ge 3$ , $ y=?$ .

# Tag: equation

## Using Cramer’s Rule to solve a system of linear equation

I have trouble solving the following question: Question

How I approached it initially was as follow:

1) Since the questions asks for x3 only, I replaced the x3 column with the particular solution. Step 1

2) Cramer’s Rule asks of us to **divide the det(Substituted Matrix in Step 1) by the det(Initial Matrix)** Step 2

3) To find the det of each 4×4 matrix I did C(1,1)(det(3×3 sub Matrix)). So 1(det(3×3 sub Matrix)). I evaluated the 3×3 sub Matrix as follow: 3×3 sub Matrix The red arrows are added and then subtracted to the addition of the blue arrows. Thus **det(Substituted Matrix) = 3aei + 3bfg + 2cdh – 3ceg – 2afh – 3bdi.**

4) I did step 3 for the initial Matrix as well and got **det(Initial Matrix) = aei + bfg + cdh – ceg – afh – bdi**

Now, I am unsure how to work around the determinants I just got. I should factor out something and then cancel the rest, but I don’t see what. Did I miss something?

## Fredholm integral equation needs to be written as a sum of functions

solve the equation

$ $ f(x) + \int_0^1 (xy+x^2y^2) f(y) dy = g(x) $ $

and write in the form of

$ $ \sum a_jx^{j-1} $ $

I have tried integration by parts but it doesn’t seem to work because of f(y).

Any assistance on the method will be much appreciated.

## Including the mean in differential equation

I have a differential equation in the form: $ $ \frac{1}{C} \frac{dC}{dt} = a – C – \mu_C $ $

where C is a random variable. Is it possible to derive an exact solution to this equation to get the time-dependence of the variable C and its distribution? I know that we can do so without the average term at the end but am not sure how to deal with the mean due to its time dependence.

## Solving equation with multi variable matrix input

Background, here are the equations that I am trying to solve:

Where R, E1, E2, V1, V2, P are all user inputs. X/A goes from -2 to 2 and Z/A goes from 0 to -2. Below is the code that I have so far. I created a list of inputs. Then created two arrays for the x and z inputs. The last is where I am having trouble. I’m trying to create a code such that it will hold a value for X constant in SX, SZ, and TXZ and plug in all the values for Z. Then move to the next value for X and plug all the values in for all the Z. The end goal is to create a density plot that for SX, SZ, and TXZ. Thank you!

`R = .1; E1 = 200*10^9; E2 = 550*10^9; P=1000; V1 = 0.3; V2 = 0.3; E = 1/(((1-(V1^2))/E1)+((1-(V2^2))/E2)); A = ((.75*P*R)/(1.61172*10^11))^(1/3); X = Range[-2 A, 2 A, 0.01*3*A]; Z = Range[0,-2 A, 0.005*3*A]; ZZ = ConstantArray[Z[[Range[Length[Z]]]], Length[X]]; XX = ConstantArray[X[[Range[Length[X]]]],Length[Z]]; For[i=1,i=Length[XX], For[j=1,j = Length[ZZ], M = Sqrt(.5*(((A^2-i^2+j^2)^2+4*i^2*j^2)^(.5)+(A^2-i^2+j^2))) N = Sqrt(.5*(((A^2-i^2+j^2)^2+4*i^2*j^2)^(.5)-(A^2-i^2+j^2))) SX = (-P/A)*M*((1-((j^2+N^2)/(M^2+N^2)))-2*N) SZ = (-P/A)*M*((1-((j^2+N^2)/(M^2+N^2)))) SY = V1*(SX+SZ) TXZ = (-P/A)*N*((M^2-j^2)/(M^2+N^2)), DensityPlot[SX/P,XX/A,ZZ/A] ] ] `