Obtain the differential equation by eliminating the arbitrary constants: c₁x³+c₂y²=5

Exercise 6.2 · Q19

c_1x^3+c_2y^2=5

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$c_1x^3+c_2y^2=5$ has two arbitrary constants, so differentiate twice. First: $3c_1x^2+2c_2y\dfrac{dy}{dx}=0$ . Differentiate again: $6c_1x+2c_2\left[\left(\dfrac{dy}{dx}\right)^2+y\dfrac{d^2y}{dx^2}\right]=0$ . From the first, $c_1=-\dfrac{2c_2y\,dy/dx}{3x^2}$ ; substitute into the second and cancel $c_2$ : $-\dfrac{4y\,dy/dx}{x}+2\left[\left(\dfrac{dy}{dx}\right)^2+y\dfrac{d^2y}{dx^2}\right]=0$ , which simplifies (mul …

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Q.Obtain the differential equation by eliminating the arbitrary constants: $c_1x^3+c_2y^2=5$

Concept understanding — Formation of a Differential Equation whose General Solution is Given

Q.Obtain the differential equation by eliminating the arbitrary constants: c1x3+c2y2=5c_1x^3+c_2y^2=5c1​x3+c2​y2=5

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Q.Obtain the differential equation by eliminating the arbitrary constants: $c_1x^3+c_2y^2=5$

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