# Differential equation with initial conditions

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#### Exotique

Find the unique function $$\displaystyle y(x)$$ satisfying the differential equation with initial condition,

$$\displaystyle \frac{dy}{dx}=x^2 y$$, $$\displaystyle y(1)=1$$

#### 11rdc11

Find the unique function $$\displaystyle y(x)$$ satisfying the differential equation with initial condition,

$$\displaystyle \frac{dy}{dx}=x^2 y$$, $$\displaystyle y(1)=1$$
$$\displaystyle \int \frac{dy}{y} = \int x^2 dx$$

$$\displaystyle \ln{|y|} = \frac{x^3}{3} + C$$

Can you take it fron here?

Exotique

#### Exotique

I'm guessing I have to solve for y in terms of x, let y = 1 and x = 1, then solve for C?

This is what I did:

$$\displaystyle y = e^\frac{x^3}{3} + C$$

Since $$\displaystyle y(1)=1$$,

$$\displaystyle 1=e^\frac{1}{3} + C$$

$$\displaystyle C=1 - e^\frac{1}{3}$$

For some reason this doesn't feel right.

Then the original equation should be:

$$\displaystyle y = e^\frac{x^3}{3} + 1 - e^\frac{1}{3}$$

Rapha

#### 11rdc11

I'm guessing I have to solve for y in terms of x, let y = 1 and x = 1, then solve for C?

This is what I did:

$$\displaystyle y = e^\frac{x^3}{3} + C$$

Since $$\displaystyle y(1)=1$$,

$$\displaystyle 1=e^\frac{1}{3} + C$$

$$\displaystyle C=1 - e^\frac{1}{3}$$

For some reason this doesn't feel right.

Then the original equation should be:

$$\displaystyle y = e^\frac{x^3}{3} + 1 - e^\frac{1}{3}$$
$$\displaystyle y= e^{\frac{x^3}{3}+ C}$$

$$\displaystyle y = e^\frac{x^3}{3}e^C$$

$$\displaystyle y=Ce^{\frac{x^3}{3}}$$

$$\displaystyle 1=Ce^{\frac{1}{3}}$$

$$\displaystyle \frac{1}{e^{\frac{1}{3}}} = C$$

Rapha and Exotique

#### Exotique

How did you get from
$$\displaystyle y=e^\frac{x^3}{3} e^C$$

to
$$\displaystyle y=Ce^\frac{x^3}{3}$$

#### pickslides

MHF Helper
$$\displaystyle e$$ to the power of an arbitrary constant remains an arbitrary constant. Therefore we can say $$\displaystyle e^C = C$$

Exotique

#### Exotique

Ahh, I understand now, thank you to all.

#### mr fantastic

MHF Hall of Fame
Ahh, I understand now, thank you to all.
This question counts towards the student's final grade. It is not MHF policy to knowingly assist with such questions. Thread closed.

Chris L T521
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