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Solve the differential equation $\left(t^2+1\right)w^{\prime}+tw=t$

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Solution

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Final answer to the problem

$w=\frac{\sqrt{t^2+1}+C_0}{\sqrt{t^2+1}}$
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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.

$\left(t^2+1\right)\frac{dw}{dt}+tw=t$

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Learn how to solve integration by trigonometric substitution problems step by step online. Solve the differential equation (t^2+1)w^'+tw=t. Rewrite the differential equation using Leibniz notation. Divide all the terms of the differential equation by t^2+1. 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(t)=\frac{t}{t^2+1} and Q(t)=\frac{t}{t^2+1}. In order to solve the differential equation, the first step is to find the integrating factor \mu(x).

Final answer to the problem

$w=\frac{\sqrt{t^2+1}+C_0}{\sqrt{t^2+1}}$

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Function Plot

Plotting: $w=\frac{\sqrt{t^2+1}+C_0}{\sqrt{t^2+1}}$

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7
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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:

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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

See formulas (4)

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