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Solve the differential equation $x\frac{dy}{dx}+y=3x\sqrt{y}$

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Solution

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Final answer to the problem

$y=\frac{\left(\sqrt{x^{3}}+C_0\right)^{2}}{x}$
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Step-by-step Solution

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equation
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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1

Rewrite the differential equation in standard form

Learn how to solve integrals of polynomial functions problems step by step online.

$\frac{dy}{dx}+\frac{y}{x}=\frac{3x\sqrt{y}}{x}$

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Unlock the first 3 steps of this solution

Learn how to solve integrals of polynomial functions problems step by step online. Solve the differential equation xdy/dx+y=3xy^(1/2). Rewrite the differential equation in standard form. Simplify the fraction \frac{3x\sqrt{y}}{x} by x. We identify that the differential equation \frac{dy}{dx}+\frac{y}{x}=3\sqrt{y} is a Bernoulli differential equation since it's of the form \frac{dy}{dx}+P(x)y=Q(x)y^n, where n is any real number different from 0 and 1. To solve this equation, we can apply the following substitution. Let's define a new variable u and set it equal to. Plug in the value of n, which equals \frac{1}{2}.

Final answer to the problem

$y=\frac{\left(\sqrt{x^{3}}+C_0\right)^{2}}{x}$

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Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

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Function Plot

Plotting: $x\frac{dy}{dx}+y-3x\sqrt{y}$

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5
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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
D□x
∫
∫◻
|◻|
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:

Similar Problems Solved

$\left(x^2+1\right)\frac{dy}{dx}+3xy=6x$

$\left(x+\frac{y}{x}\right)dx-dy=0$

$x\frac{dy}{dx}+2y=3$

$\frac{dy}{dx}=\frac{x^3-2y}{x}$

$x^3\frac{dy}{dx}+3x^2y=x$

$x\frac{dy}{dx}+2y=x^{-3}$

$dy=\left(x-3y\right)dx$

Main Topic: Integrals of Polynomial Functions

Integrals of polynomial functions.

Used Formulas

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