Find the general solution to the differential equation.dydx=yx\frac{dy}{dx}=y\sqrt{x}

y = C e 2 3 x 3 2 y=Ce^{\frac23x^{\frac32}}

Find the general solution to the differential equation. d y d x = y x \frac{dy}{dx}=y\sqrt{x}

this problem, we're asked to find the general solution to the differential equation dy dx equals y times the square root of x. Now looking at this equation, we can clearly see that this is separable as it's already explicitly written in the form that dy dx is equal to some function of x multiplying some function of y. So we wanna go ahead and get started with our step steps here where the first thing we wanna do is separate our variables. So if d y d x is equal to y times the square root of x, we want to get that function of y and d y together on one side, our function of x and d x together on the other. So the first thing I'm going to do here is divide both sides by y. Remember, that's the same thing as multiplying by one over y. So now I have moved that over, and I can go ahead and move that d x over in the same step as I multiply both sides by d x in order to cancel that out on the left. So now I have one over y dy equals the square root of x times dx. So with those variables separated, I can move on to my next step, which is integrating both sides. So integrating each side here, the integral of one over y d y is going to be the natural log of the absolute value of y, and that's then equal to the integral of the square root of x d x. We can apply our power rule here remember the square root of x is the same thing as x to the power of one half. So So this is gonna give us here two thirds x to the power of three halves, and then plus our constant of integration c. So we finished that step two. And now since we're asked to find the general solution, we can go ahead and move on to step four, skipping over that step three since we're not given an initial condition with which we could find c. So now we're just solving for y. So I wanna get this y all by itself, and I can do that by taking e and raising it to each side. So that e and natural log will cancel out, making this the absolute value of y, and that's then equal to e to the power of two thirds, x to the power of three halves, and then plus c. Now you'll see a couple of things happen here that may get skipped whenever you watch other solutions, and that's because oftentimes, tutors will do this all in one step. But I'm gonna go ahead and break this down for you. There are a couple of things that we wanna change. We wanna get this c in front of that e by rewriting this using exponential rules. So I can rewrite this as e to the power of two thirds, x to the power of three halves times e to the power of c. Now this all by itself is just another constant, so we could effectively just call this c. Now you may see people refer to this as a different constant, say c one, but it doesn't matter because in the end, it all just ends up being a constant. So I'm gonna go ahead and keep that as c. So now I can rewrite this as c times e to the power of two thirds, x to the power of three halves. And this is equal to the absolute value of y. Now you may be wondering here how we get rid of these absolute value signs and why oftentimes they seem to just get ignored. And this is because in the same step, we can go ahead and rewrite this right hand side as a plus or minus c times e to the power of two thirds x to the power of three halves. But oftentimes we're just gonna absorb that plus or minus altogether in that c because no matter if it's positive or negative it's still a constant. So we can really just write this as y equals c times e to the power of two thirds x to the power of three halves, and this is our final answer. Now everything that I did in these final couple of steps is something that you may often see get skipped, but just know that that's really what's happening here as we get to our final answer. Let us know if you have any questions, and I'll see you in the next one.

Find the general solution to the differential equation. d y d x = y x \frac{dy}{dx}=y\sqrt{x}

y = C e 2 3 x 3 2 y=Ce^{\frac23x^{\frac32}}

y = 2 3 x 3 2 + C y=\frac23x^{\frac32}+C

y = 3 2 x 3 2 + C y=\frac32x^{\frac32}+C

y = C e 2 3 x 2 3 y=Ce^{\frac23x^{\frac23}}

13. Intro to Differential Equations

Separable Differential Equations

Calculus

13. Intro to Differential Equations

Separable Differential Equations

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Find the general solution to the differential equation.
$\frac{d y}{d x} = y \sqrt{x}$

$y = \frac{2}{3} x^{\frac{3}{2}} + C$

$y = \frac{3}{2} x^{\frac{3}{2}} + C$

$y = C e^{\frac{2}{3} x^{\frac{3}{2}}}$

$y = C e^{\frac{2}{3} x^{\frac{2}{3}}}$

Verified step by step guidance

Step 1: Start by identifying the type of differential equation. The given equation $ \frac{dy}{dx} = y \sqrt{x} $ is a first-order separable differential equation because the variables $ y $ and $ x $ can be separated.

Step 2: Rewrite the equation to separate the variables. Divide both sides by $ y $ and multiply by $ dx $: $ \frac{1}{y} dy = \sqrt{x} dx $.

Step 3: Integrate both sides. The left-hand side integrates to $ \ln|y| $, and the right-hand side integrates to $ \frac{2}{3}x^{\frac{3}{2}} + C $, where $ C $ is the constant of integration.

Step 4: Solve for $ y $ by exponentiating both sides to remove the natural logarithm. This gives $ y = Ce^{\frac{2}{3}x^{\frac{3}{2}}} $, where $ C $ is a positive constant (absorbing the absolute value).

Step 5: Verify the solution by substituting $ y = Ce^{\frac{2}{3}x^{\frac{3}{2}}} $ back into the original differential equation $ \frac{dy}{dx} = y \sqrt{x} $ to ensure it satisfies the equation.

7:15m

Watch next

Master Separation of Variables with a bite sized video explanation from Patrick

Start learning

Separable Differential Equations practice set

Problem sets built by lead tutors

Expert video explanations

Go to Practice