Solved on Feb 01, 2024

Solve the differential equation $\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2 \sqrt{y}$ .

STEP 1

Assumptions
1. The given differential equation is separable, meaning we can express it in the form $f(y)dy = g(x)dx$ .
2. The function $y(x)$ is differentiable, and its derivative with respect to $x$ is given by $\frac{dy}{dx}$ .
3. We will integrate both sides of the equation to find the general solution.

STEP 2

Rewrite the differential equation to separate the variables $y$ and $x$ .
$\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2(\sqrt{y})$

STEP 3

Separate the variables by multiplying both sides by $dx$ and dividing by $\sqrt{y} \cos^2(\sqrt{y})$ .
$\frac{dy}{\sqrt{y} \cos^2(\sqrt{y})} = \frac{1}{5} dx$

STEP 4

Integrate both sides of the equation with respect to their respective variables.
$\int \frac{1}{\sqrt{y} \cos^2(\sqrt{y})} dy = \int \frac{1}{5} dx$

STEP 5

Let's make a substitution to simplify the left-hand side integral. Set $u = \sqrt{y}$ , which implies $y = u^2$ and $dy = 2u du$ .

STEP 6

Substitute $u$ and $dy$ into the left-hand side integral.
$\int \frac{1}{u \cos^2(u)} 2u du = \int \frac{1}{5} dx$

STEP 7

Simplify the left-hand side integral.
$\int \frac{2}{\cos^2(u)} du = \int \frac{1}{5} dx$

STEP 8

Recognize that $\frac{2}{\cos^2(u)}$ is $2\sec^2(u)$ , where $\sec(u) = \frac{1}{\cos(u)}$ .
$\int 2\sec^2(u) du = \int \frac{1}{5} dx$

STEP 9

Integrate both sides. The integral of $2\sec^2(u)$ with respect to $u$ is $2\tan(u)$ , and the integral of $\frac{1}{5}$ with respect to $x$ is $\frac{x}{5}$ .
$2\tan(u) = \frac{x}{5} + C$

STEP 10

Now, substitute back $u = \sqrt{y}$ to express the solution in terms of $y$ .
$2\tan(\sqrt{y}) = \frac{x}{5} + C$

STEP 11

To express $y$ explicitly in terms of $x$ , we would need to take the inverse tangent of both sides and then square the result. However, the constant $C$ complicates this process, and it's not always possible to express $y$ in a simple closed form. Therefore, we leave the solution in its implicit form.
The general solution to the differential equation is:
$2\tan(\sqrt{y}) = \frac{x}{5} + C$

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Solve the differential equation $\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2 \sqrt{y}$ .

STEP 1

STEP 2

Rewrite the differential equation to separate the variables $y$ and $x$ .
$\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2(\sqrt{y})$

STEP 3

Separate the variables by multiplying both sides by $dx$ and dividing by $\sqrt{y} \cos^2(\sqrt{y})$ .
$\frac{dy}{\sqrt{y} \cos^2(\sqrt{y})} = \frac{1}{5} dx$

STEP 4

Integrate both sides of the equation with respect to their respective variables.
$\int \frac{1}{\sqrt{y} \cos^2(\sqrt{y})} dy = \int \frac{1}{5} dx$

STEP 5

Let's make a substitution to simplify the left-hand side integral. Set $u = \sqrt{y}$ , which implies $y = u^2$ and $dy = 2u du$ .

STEP 6

Substitute $u$ and $dy$ into the left-hand side integral.
$\int \frac{1}{u \cos^2(u)} 2u du = \int \frac{1}{5} dx$

STEP 7

Simplify the left-hand side integral.
$\int \frac{2}{\cos^2(u)} du = \int \frac{1}{5} dx$

STEP 8

Recognize that $\frac{2}{\cos^2(u)}$ is $2\sec^2(u)$ , where $\sec(u) = \frac{1}{\cos(u)}$ .
$\int 2\sec^2(u) du = \int \frac{1}{5} dx$

STEP 9

Integrate both sides. The integral of $2\sec^2(u)$ with respect to $u$ is $2\tan(u)$ , and the integral of $\frac{1}{5}$ with respect to $x$ is $\frac{x}{5}$ .
$2\tan(u) = \frac{x}{5} + C$

STEP 10

Now, substitute back $u = \sqrt{y}$ to express the solution in terms of $y$ .
$2\tan(\sqrt{y}) = \frac{x}{5} + C$

STEP 11

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Solve the differential equation dydx=15ycos⁡2y\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2 \sqrt{y}dxdy​=51​y​cos2y​.

STEP 1

STEP 2

Rewrite the differential equation to separate the variables y y y and x x x.dydx=15ycos⁡2(y) \frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2(\sqrt{y}) dxdy​=51​y​cos2(y​)

STEP 3

Separate the variables by multiplying both sides by dx dx dx and dividing by ycos⁡2(y) \sqrt{y} \cos^2(\sqrt{y}) y​cos2(y​).dyycos⁡2(y)=15dx \frac{dy}{\sqrt{y} \cos^2(\sqrt{y})} = \frac{1}{5} dx y​cos2(y​)dy​=51​dx

STEP 4

Integrate both sides of the equation with respect to their respective variables.∫1ycos⁡2(y)dy=∫15dx \int \frac{1}{\sqrt{y} \cos^2(\sqrt{y})} dy = \int \frac{1}{5} dx ∫y​cos2(y​)1​dy=∫51​dx

STEP 5

Let's make a substitution to simplify the left-hand side integral. Set u=y u = \sqrt{y} u=y​, which implies y=u2 y = u^2 y=u2 and dy=2udu dy = 2u du dy=2udu.

STEP 6

Substitute u u u and dy dy dy into the left-hand side integral.∫1ucos⁡2(u)2udu=∫15dx \int \frac{1}{u \cos^2(u)} 2u du = \int \frac{1}{5} dx ∫ucos2(u)1​2udu=∫51​dx

STEP 7

Simplify the left-hand side integral.∫2cos⁡2(u)du=∫15dx \int \frac{2}{\cos^2(u)} du = \int \frac{1}{5} dx ∫cos2(u)2​du=∫51​dx

STEP 8

Recognize that 2cos⁡2(u) \frac{2}{\cos^2(u)} cos2(u)2​ is 2sec⁡2(u) 2\sec^2(u) 2sec2(u), where sec⁡(u)=1cos⁡(u) \sec(u) = \frac{1}{\cos(u)} sec(u)=cos(u)1​.∫2sec⁡2(u)du=∫15dx \int 2\sec^2(u) du = \int \frac{1}{5} dx ∫2sec2(u)du=∫51​dx

STEP 9

Integrate both sides. The integral of 2sec⁡2(u) 2\sec^2(u) 2sec2(u) with respect to u u u is 2tan⁡(u) 2\tan(u) 2tan(u), and the integral of 15 \frac{1}{5} 51​ with respect to x x x is x5 \frac{x}{5} 5x​.2tan⁡(u)=x5+C 2\tan(u) = \frac{x}{5} + C 2tan(u)=5x​+C

STEP 10

Now, substitute back u=y u = \sqrt{y} u=y​ to express the solution in terms of y y y.2tan⁡(y)=x5+C 2\tan(\sqrt{y}) = \frac{x}{5} + C 2tan(y​)=5x​+C

STEP 11

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Solve the differential equation $\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2 \sqrt{y}$ .

Rewrite the differential equation to separate the variables $y$ and $x$ .
$\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2(\sqrt{y})$

Separate the variables by multiplying both sides by $dx$ and dividing by $\sqrt{y} \cos^2(\sqrt{y})$ .
$\frac{dy}{\sqrt{y} \cos^2(\sqrt{y})} = \frac{1}{5} dx$

Integrate both sides of the equation with respect to their respective variables.
$\int \frac{1}{\sqrt{y} \cos^2(\sqrt{y})} dy = \int \frac{1}{5} dx$

Let's make a substitution to simplify the left-hand side integral. Set $u = \sqrt{y}$ , which implies $y = u^2$ and $dy = 2u du$ .

Substitute $u$ and $dy$ into the left-hand side integral.
$\int \frac{1}{u \cos^2(u)} 2u du = \int \frac{1}{5} dx$

Simplify the left-hand side integral.
$\int \frac{2}{\cos^2(u)} du = \int \frac{1}{5} dx$

Recognize that $\frac{2}{\cos^2(u)}$ is $2\sec^2(u)$ , where $\sec(u) = \frac{1}{\cos(u)}$ .
$\int 2\sec^2(u) du = \int \frac{1}{5} dx$

Integrate both sides. The integral of $2\sec^2(u)$ with respect to $u$ is $2\tan(u)$ , and the integral of $\frac{1}{5}$ with respect to $x$ is $\frac{x}{5}$ .
$2\tan(u) = \frac{x}{5} + C$

Now, substitute back $u = \sqrt{y}$ to express the solution in terms of $y$ .
$2\tan(\sqrt{y}) = \frac{x}{5} + C$