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Solve the differential equation $\frac{yx}{\sqrt{1+x^2}\left(1-\sqrt{1+x^2}\right)}=\frac{dy}{dx}$

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Solution

Final answer to the problem

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

Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side of the equality

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$y\cdot dy=\frac{\sqrt{1+x^2}\left(1-\sqrt{1+x^2}\right)}{x}dx$

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Learn how to solve problems step by step online. Solve the differential equation (yx)/((1+x^2)^(1/2)(1-(1+x^2)^(1/2)))=dy/dx. Group the terms of the differential equation. Move the terms of the y variable to the left side, and the terms of the x variable to the right side of the equality. Integrate both sides of the differential equation, the left side with respect to y, and the right side with respect to x. Solve the integral \int ydy and replace the result in the differential equation. Solve the integral \int\frac{\sqrt{1+x^2}\left(1-\sqrt{1+x^2}\right)}{x}dx and replace the result in the differential equation.

Final answer to the problem

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

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Function Plot

Plotting: $\frac{yx}{\sqrt{1+x^2}\left(1-\sqrt{1+x^2}\right)}-\frac{dy}{dx}$

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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
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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