How do you solve for the equation dydx=3x2e2y that satisfies the initial condition f(0)=12?

First of all, I think ther is a mistake in your writing, I think you wanted to write:dydx=3x2e2yThis is a separable differential equations, so:e2ydy=3x2dx⇒∫e2ydy=∫3x2dx⇒12e2y=x3+cNow to find c let's use the condition: f(0)=12e.So the solution is:12e2y=x3+12e⇒e2y=2x3+e⇒2y=ln⁡(2x3+e)⇒y=12ln⁡(2x3+e)

Check the solution to "How do you solve for the equation dydx=3x2e2y..."

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Malik Dean

Answered question

2022-04-10

How do you solve for the equation

\frac{d y}{d x} = \frac{3 x^{2}}{e^{2} y}

that satisfies the initial condition

f (0) = \frac{1}{2}

Answer & Explanation

ysnlm8eut

Beginner2022-04-11Added 14 answers

First of all, I think ther is a mistake in your writing, I think you wanted to write:

\frac{d y}{d x} = \frac{3 x^{2}}{e^{2 y}}

This is a separable differential equations, so:

e^{2 y} d y = 3 x^{2} d x \Rightarrow \int e^{2 y} d y = \int 3 x^{2} d x \Rightarrow

\frac{1}{2} e^{2 y} = x^{3} + c

Now to find c let's use the condition:

f (0) = \frac{1}{2} e .

So the solution is:

\frac{1}{2} e^{2 y} = x^{3} + \frac{1}{2} e \Rightarrow e^{2 y} = 2 x^{3} + e \Rightarrow 2 y = \ln (2 x^{3} + e) \Rightarrow

y = \frac{1}{2} \ln (2 x^{3} + e)

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 2

What is the derivative of the work function?
How to use implicit differentiation to find $\frac{d y}{d x}$ given $3 x^{2} + 3 y^{2} = 2$ ?
How to differentiate $y = \log x^{2}$ ?
The solution of a differential equation y′′+3y′+2y=0 is of the form
A) $c_{1} e^{x} + c_{2} e^{2 x}$
B) $c_{1} e^{- x} + c_{2} e^{3 x}$
C) $c_{1} e^{- x} + c_{2} e^{- 2 x}$
D) $c_{1} e^{- 2 x} + c_{2} 2^{- x}$
How to find instantaneous velocity from a position vs. time graph?
How to implicitly differentiate $\sqrt{x y} = x - 2 y$ ?
What is 2xy differentiated implicitly?
How to find the sum of the infinite geometric series given $1 + \frac{2}{3} + \frac{4}{9} + ...$ ?
Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?
A. 4.2
B. 4.4
C. 4.7
D. 5.1
What is the derivative of $\frac{x + 1}{y}$ ?
How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?
How to find the volume of a cone using an integral?
What is the surface area of the solid created by revolving $f (x) = e^{2 - x}, x \in [1, 2]$ around the x axis?
How to differentiate $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 4$ ?
The differential coefficient of $\sec (\tan^{- 1} (x))$ .

Didn't find what you were looking for?