Topics

More AI tools from SnapXam

Exercise

$\frac{dp}{dt}=-\frac{3t}{\sqrt{p}}$

Step-by-step Solution

1

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

$\sqrt{p}dp=-3tdt$
2

Integrate both sides of the differential equation, the left side with respect to $p$, and the right side with respect to $t$

$\int\sqrt{p}dp=\int-3tdt$
3

Solve the integral $\int\sqrt{p}dp$ and replace the result in the differential equation

$\frac{2\sqrt{p^{3}}}{3}=\int-3tdt$
4

Solve the integral $\int-3tdt$ and replace the result in the differential equation

$\frac{2\sqrt{p^{3}}}{3}=-\frac{3}{2}t^2+C_0$
5

Find the explicit solution to the differential equation. We need to isolate the variable $p$

$p=\frac{\sqrt[3]{\left(\frac{3\left(-3t^2+C_1\right)}{2}\right)^{2}}}{\sqrt[3]{\left(2\right)^{2}}}$

Final answer to the exercise

$p=\frac{\sqrt[3]{\left(\frac{3\left(-3t^2+C_1\right)}{2}\right)^{2}}}{\sqrt[3]{\left(2\right)^{2}}}$

Try other ways to solve this exercise

  • Choose an option
  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equations
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
  • Load more...
Can't find a method? Tell us so we can add it.

Other solved exercises

$\lim_{x\to\infty}\left(\frac{2x^3-2x^2+x-5}{x^3+2x^2-x+4}\right)$

$2x^3-6x-4$

$2-7x\le16$

$2\left(x^2+5x\right)+2\left(y^2+9y\right)$

$2\cos^2\left(x\right)-2\sin^2\left(x\right)=2-4\sin^2\left(x\right)$

$m^2+7m^2w^5+49w^1^0$

$8x\left(-8\right)$
Symbolic mode
Text mode
Go!
1
2
3
4
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

Empowering students and teachers through AI

By signing up, you agree to SnapXam's Terms and Privacy Policy.

Already have an account? Login here
Forgot your password?

Empowering students and teachers through AI



Register here if you don't have an account.
Forgot your password?

Your Personal Math Tutor. Powered by AI

Available 24/7, 365 days a year.

Complete step-by-step math solutions. No ads.

Choose between multiple solving methods.

Download unlimited solutions in PDF format.

Premium access on our iOS and Android apps.

Join 1M+ students worldwide in problem solving.

Choose the plan that suits you best:
Pay $39.97 USD securely with your payment method.
Please hold while your payment is being processed.
Create an Account