Find a general solution y(x) to the differential equation y′=yx+2x2

Gregory Emery

Gregory Emery

Answered

2021-12-23

Find a general solution y(x) to the differential equation y=yx+2x2

Answer & Explanation

Jimmy Macias

Jimmy Macias

Expert

2021-12-24Added 30 answers

dydx=yx+2x2
dydxyx=2x2
(check linear form of differential equation)
P=1x, Q=2x2
IF=epdx=e1xdx
=elogx=x
y(IF)=Q(IF)+C
yx=2x3dx+C
xy=x42+C
y=x32+cx - is the solution
William Appel

William Appel

Expert

2021-12-25Added 44 answers

Another solution?
user_27qwe

user_27qwe

Expert

2021-12-30Added 230 answers

Yes, here:
Solve the linear equation (dy(x))/(dx)=2x2+y(x)/x:
Subtract y(x)/x from both sides:
(dy(x))/(dx)y(x)/x=2x2
Let μ(x)=e(int1/xdx)=1/x
Multiply both sides by μ(x):
((dy(x))/(dx))/xy(x)/x2=2x
Substitute 1/x2=d/(dx)(1/x):
((dy(x))/(dx))/x+d/(dx)(1/x)y(x)=2x
Apply the reverse product rule f(dg)/(dx)+g(df)/(dx)=d/(dx)(fg) to the left-hand side:
d/(dx)(y(x)/x)=2x
Integrate both sides with respect to x:
d/(dx)(y(x)/x)dx=2xdx
Evaluate the integrals:
y(x)/x=x2+c1, where c1 is an arbitrary constant.
Divide both sides by μ(x)=1/x
Answer:
y(x)=x(x2+c1)

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