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Solve the differential equation $\left(x^3y+y^4\right)dx-x^4dy=0$

Step-by-step Solution

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Final Answer

$y=\frac{x}{\sqrt[3]{-3\ln\left(x\right)+C_2}}$
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Step-by-step Solution

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We can identify that the differential equation $\left(x^3y+y^4\right)dx-x^4dy=0$ is homogeneous, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and both are homogeneous functions of the same degree

$\left(x^3y+y^4\right)dx-x^4dy=0$

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$\left(x^3y+y^4\right)dx-x^4dy=0$

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Learn how to solve differential equations problems step by step online. Solve the differential equation (x^3y+y^4)dx-x^4dy=0. We can identify that the differential equation \left(x^3y+y^4\right)dx-x^4dy=0 is homogeneous, since it is written in the standard form M(x,y)dx+N(x,y)dy=0, where M(x,y) and N(x,y) are the partial derivatives of a two-variable function f(x,y) and both are homogeneous functions of the same degree. Use the substitution: y=ux. Expand and simplify. Integrate both sides of the differential equation, the left side with respect to u, and the right side with respect to x.

Final Answer

$y=\frac{x}{\sqrt[3]{-3\ln\left(x\right)+C_2}}$

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Linear Differential EquationExact Differential EquationSeparable Differential EquationHomogeneous Differential Equation

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Plotting: $\left(x^3y+y^4\right)dx-x^4dy$

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1
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5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Main Topic: Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives.

Used Formulas

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