Question

Solve the differential equation.

y’ = x+y

Answer 1

Answer:

x=0

Step-by-step explanation:

Let's solve for x.

y=x+y

Step 1: Flip the equation.

x+y=y

Step 2: Add -y to both sides.

x+y+−y=y+−y

x=0

Answer:

x=0

Answer 2

Answer:

y′=x+y

y′=x+y

Then we let u=x+yu=x+y

This gives u′=1+y′u′=1+y′, so that the equation becomes

u′−1=u

u′−1=u

u′−u=1

u′−u=1

Can you solve that for uu?

Hint (ex−1)′=ex(ex−1)′=ex

Moving on with the solution:

dudx−u=1

dudx−u=1

dudx=1+u

dudx=1+u

And the classic abuse in DE's

duu+1=dx

duu+1=dx

Now

∫duu+1=∫dx

∫duu+1=∫dx

log(u+1)=x+C

log⁡(u+1)=x+C

We take logarithms

u+1=ex+C

u+1=ex+C

We use the property of the exponential function f(x+y)=f(x)f(y)f(x+y)=f(x)f(y)

u+1=eCex

u+1=eCex

Here K=eCK=eC

y+x+1=Kex

y+x+1=Kex

y=Kex−x−1