Final answer to the problem
$-\frac{1}{2}\ln\left(\frac{y}{x}+2\right)+\frac{1}{2}\ln\left(\frac{y}{x}+1\right)=\ln\left(x\right)+C_0$
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Step-by-step Solution
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- Exact Differential Equation
- Linear Differential Equation
- Separable Differential Equation
- 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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1
Rewrite the differential equation using Leibniz notation
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$x^2\frac{dy}{dx}=4x^2+7xy+2y^2$
Learn how to solve problems step by step online. Solve the differential equation x^2y^'=4x^2+7xy2y^2. Rewrite the differential equation using Leibniz notation. Rewrite the differential equation. We can identify that the differential equation \frac{dy}{dx}=\frac{4x^2+7xy+2y^2}{x^2} is homogeneous, since it is written in the standard form \frac{dy}{dx}=\frac{M(x,y)}{N(x,y)}, where M(x,y) and N(x,y) are the partial derivatives of a two-variable function f(x,y) and both are homogeneous functions of the same degree. Use the substitution: y=ux.
Final answer to the problem
$-\frac{1}{2}\ln\left(\frac{y}{x}+2\right)+\frac{1}{2}\ln\left(\frac{y}{x}+1\right)=\ln\left(x\right)+C_0$
