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Exercise

$\frac{dy}{dx}=\left(3\sqrt{y+1}\right)\cdot\cos\left(x\right)$

Step-by-step Solution

1

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

$\frac{1}{\sqrt{y+1}}dy=3\cos\left(x\right)dx$
2

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

$\int\frac{1}{\sqrt{y+1}}dy=\int3\cos\left(x\right)dx$
3

Solve the integral $\int\frac{1}{\sqrt{y+1}}dy$ and replace the result in the differential equation

$2\sqrt{y+1}=\int3\cos\left(x\right)dx$
4

Solve the integral $\int3\cos\left(x\right)dx$ and replace the result in the differential equation

$2\sqrt{y+1}=3\sin\left(x\right)+C_0$

Final answer to the exercise

$2\sqrt{y+1}=3\sin\left(x\right)+C_0$

Try other ways to solve this exercise

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  • 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
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Similar Questions Solved

$\frac{dy}{dx}=\tan\left(x+y\right)$

$ydy=4x\left(y^2+1\right)^{\frac{1}{2}}dx$

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Main Topic: Separable Differential Equations

A separable differential equation is one that can be solved by separating variables, that is, when all expressions involving a variable can be placed on one side of the equation, and expressions involving the other variable on the other side of the equation.

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