First-Order Linear Differential Equations; Solutions Suggested by the Equation dydx=(x+y+1)2

Daniell Phillips

Daniell Phillips

Answered

2021-12-26

First-Order Linear Differential Equations; Solutions Suggested by the Equation
dydx=(x+y+1)2

Answer & Explanation

Hector Roberts

Hector Roberts

Expert

2021-12-27Added 31 answers

Step 1
We have given the differential equation as dydx=(x+y+1)2. To solve this differential equation, we have to use substitution to convert the equation into the separable differential equation because separable equations are easy to solve.
Step 2
So substitute x+y+1=v  dydx=(x+y+1)2. First, differentiate x+y+1=v with respect to x to find the expression for dydx.
ddx(x+y+1)=v
1+dydx=dydx
dydx=dvdx1
Now, do substitution and simplify the differential equation.
dvdx1=v2
dvdx=1+v2
dv1+v2=dx
Step 3
Now, integrate both sides of dv1+v2=dx and evaluate the integral. Use the formula du1+u2=tan1u+C.
dv1+v2=dx
tan1v=x+C
tan1(x+y+1)=x+C [Substitute back x+y+1=v]
Hence, the solution of the given differential equation is tan1(x+y+1)=x+C, where C is an integral constant.
sirpsta3u

sirpsta3u

Expert

2021-12-28Added 42 answers

dydx=(x+y+1)2 (1)
Let. x+y+1=t
1+dydx=dtdx
dydx=dtdx1 (2)
from eqn(1) and (2)
dtdx1=t2
dtdx=1+t2
1.dt1+t2=dx
tan1(t)=x+C
t=tan(x+C)
x+y+1=tan(x+C)
y=tan(x+C)x1. Answer.
Vasquez

Vasquez

Expert

2022-01-09Added 457 answers

The given differential equation is
dydx=(x+y+1)2 ...(i)
Let x+y+1=v
dydx=dvdx1
dvdx1=v2
dvv2+1=dx
Integrating, we get
dvv2+1=dx
tan1(v)=x+c
tan1(x+y+1)=x+c
Which is the required solution.

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