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

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Solution

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Final answer to the problem

$\frac{3}{2}\ln\left|\frac{\left(\frac{y}{x}+1\right)x}{y-x}\right|-\frac{1}{2}\ln\left|1+\frac{-y^2}{x^2}\right|=\ln\left|x\right|+C_0$
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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 integration by trigonometric substitution problems step by step online.

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

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Unlock the first 3 steps of this solution

Learn how to solve integration by trigonometric substitution problems step by step online. Solve the differential equation y^'=(x+3y)/(3x+y). Rewrite the differential equation using Leibniz notation. We can identify that the differential equation \frac{dy}{dx}=\frac{x+3y}{3x+y} is homogeneous, since it is written in the standard form \frac{dy}{dx}=\frac{M(x,y)}{N(x,y)}, 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.

Final answer to the problem

$\frac{3}{2}\ln\left|\frac{\left(\frac{y}{x}+1\right)x}{y-x}\right|-\frac{1}{2}\ln\left|1+\frac{-y^2}{x^2}\right|=\ln\left|x\right|+C_0$

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Function Plot

Plotting: $\frac{3}{2}\ln\left(\frac{\left(\frac{y}{x}+1\right)x}{y-x}\right)-\frac{1}{2}\ln\left(1+\frac{-y^2}{x^2}\right)=\ln\left(x\right)+C_0$

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a
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f
g
m
n
u
v
w
x
y
z
.
(◻)
+
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×
◻/◻
/
÷
◻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

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Similar Problems Solved

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Main Topic: Integration by Trigonometric Substitution

Trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing radical expressions.

Used Formulas

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