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First Order Differential Equations

Differential Equations - Variable Separable

What is the solution of the following differential equation: $\left(\frac{dy}{dx}\right)^{5}-x\frac{dy}{dx}+y=0?$

Note: $C$ is a constant.

y=x-C

y=Cx-C^{5}

y=Cx-C^{4}

y=Cx

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by Brilliant Staff

What is the solution of the differential equation $\frac{1}{5}\left(\frac{1}{{x}^{3}}\frac{dx}{dy}-\frac{1}{{x}^{2}}y\right) =y\sin{{y}^{2}} ?$

Details and assumptions

Use $C$ as the constant of integration.

5=2{x}^{2}\left(\cos{{y}^{2}}-\sin{{y}^{2}}-C{e}^{-{y}^{2}}\right)

2={x}^{2}\left(\cos{{y}^{2}}-\sin{{y}^{2}}-C{e}^{-{y}^{2}}\right)

2=5{x}^{2}\left(\sin{{y}^{2}}-\cos{{y}^{2}}-C{e}^{-{y}^{2}}\right)

2=5{x}^{2}\left(\cos{{y}^{2}}-\sin{{y}^{2}}-C{e}^{-{y}^{2}}\right)

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by Brilliant Staff

What is the solution of the differential equation $\begin{aligned} (8x^2-5yx^2)\frac{dy}{dx}+(5y^3+xy^3) &=0, \\ y(1) &=1? \end{aligned}$

-\frac{4}{y^2}+\frac{10}{y}-\frac{15}{x}+\ln x+4=0

-\frac{4}{y^2}+\frac{5}{y}-\frac{5}{x}+\ln x+4=0

\frac{8}{y^2}-\frac{5}{y}+\frac{5}{x}+\ln x+4=0

\frac{8}{y^2}-\frac{10}{y}+\frac{15}{x}+\ln x+4=0

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by Brilliant Staff

What is the solution of the differential equation $\begin{aligned} \log{\frac{dy}{dx}} &=8x+7y,\\ y(0)&=0? \end{aligned}$

Note: In this problem, $\log(x)$ is the natural logarithm (log base e) of $x.$

7{e}^{8x}+8{e}^{-7y}=15

8{e}^{7x}+7{e}^{-8y}=15

7{e}^{x}+8{e}^{-y}=15

8{e}^{8x}+7{e}^{-7y}=15

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by Brilliant Staff

What is the solution of the differential equation $\log{\frac{dy}{dx}}=5x+2y, y(0)=0?$

5e^{-2y}+2e^{5x}=7

5e^{-5y}+5e^{5x}=7

5e^{-5y}+2e^{2x}=7

2e^{-2y}+2e^{2x}=7

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by Brilliant Staff