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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

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

Learn how to solve limits by direct substitution problems step by step online.

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

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Learn how to solve limits by direct 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

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Main Topic: Limits by Direct Substitution

Find limits of functions at a specific point by directly plugging the value into the function.

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