Differential equations questions and answers

Recent questions in Differential Equations
Zane Decker 2022-04-14 Answered

Laplace transform of f(t)=tetsin(2t)

Jaiden Bowman 2022-04-10 Answered

Problem:
Solve the following differential equations.
x d y d x + y = y 2
Answer:
To solve this equation, we reduce it to a linear differential equation with the substitution v = y 3 .
x y 2 d y d x + y 3 = 1 d v d x = 3 y 2 d y d x 3 x y 2 d y d x + 3 y 3 = 3 d v d x + 3 v = 3
Now we have a first order linear differential equation. To solve it, we use the integrating factor I = e P ( x ) . In this case, we have P(x)=3.
I ( x ) = e 3 x e 3 x d v d x + 3 e 3 x v = 3 e 3 x D ( e 3 x v ) = 3 e 3 x e 3 x v = e 3 x + C v = 1 + e 3 x y 3 = 1 + e 3 x
Now to check my answer.
3 y 2 d y d x = 3 C e 3 x y 2 d y d x = C e 3 x d y d x = C e 3 x y 2 x d y d x + y = x ( C e 3 x y 2 ) + ( 1 + e 3 x ) 1 3
I cannot seem to get my answer to check. Where did I go wrong?
Here is my second attempt to solve the problem:
To solve this equation, we reduce it to a linear differential equation with the substitution v = y 3 .
x y 2 d y d x + y 3 = 1 d v d x = 3 y 2 d y d x 3 x y 2 d y d x + 3 y 3 = 3 x d v d x + 3 v = 3 d v d x + 3 x 1 v = 3 x 1
Now we have a first order linear differential equation. To solve it, we use the integrating factor I = e P ( x ) . In this case, we have P ( x ) = 3 x 1 .
I = e 3 x 1 d x = e 3 ln | x | I = 3 x 3 x d v d x + 9 v = 9
Now, I want to write:
D ( 3 x v ) = 9
but that is wrong. What did I do wrong?
Here is my third attempt to solve the problem. Last time, I made a mistake in finding the integrating factor.
To solve this equation, we reduce it to a linear differential equation with the substitution v = y 3
x y 2 d y d x + y 3 = 1 d v d x = 3 y 2 d y d x 3 x y 2 d y d x + 3 y 3 = 3 x d v d x + 3 v = 3 d v d x + 3 x 1 v = 3 x 1
Now we have a first order linear differential equation. To solve it, we use the integrating factor I = e P ( x ) . In this case, we have P ( x ) = 3 x 1 .
I = e 3 x 1 d x = e 3 ln | x | I = x 3 x 3 d v d x + 3 x 2 v = 3 x 2 D ( x 3 v ) = x 3 + C x 3 v = x 3 + C v = C x 3 + 1 y 3 = C x 3 + 1
Do I have it right now?

2022-04-03

Speaking of differential equations, these are used not only by those students majoring in Physics because solving differential equations is also quite common in Statistics and Financial Studies. Explore the list of questions and examples of equations to get a basic idea of how it is done.

These answers below are meant to provide you with the starting points as you work with your differential equations. If you need specific help or cannot understand the rules behind the answers that are presented below, start with a simple equation and learn with the provided solutions..