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Solve the differential equation $xy^{\prime}+y=x+x^3$

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Solution

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Final answer to the problem

$xy=\frac{1}{2}x^2+\frac{x^{4}}{4}+C_0$
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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 using Leibniz notation

Learn how to solve differential equations problems step by step online.

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

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Unlock the first 3 steps of this solution

Learn how to solve differential equations problems step by step online. Solve the differential equation xy^'+y=x+x^3. Rewrite the differential equation using Leibniz notation. Divide all the terms of the differential equation by x. Simplifying. We can identify that the differential equation has the form: \frac{dy}{dx} + P(x)\cdot y(x) = Q(x), so we can classify it as a linear first order differential equation, where P(x)=\frac{1}{x} and Q(x)=\frac{x+x^3}{x}. In order to solve the differential equation, the first step is to find the integrating factor \mu(x).

Final answer to the problem

$xy=\frac{1}{2}x^2+\frac{x^{4}}{4}+C_0$

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Function Plot

Plotting: $xy=\frac{1}{2}x^2+\frac{x^{4}}{4}+C_0$

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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: Differential Equations

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

Used Formulas

See formulas (2)

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