MHB Solving a Separable Variable Differential Equation Using U-Substitution

  • Thread starter Thread starter karush
  • Start date Start date
karush
Gold Member
MHB
Messages
3,240
Reaction score
5
Solve the de $$\dfrac{dy}{dx}=\dfrac{1}{7}\sqrt{y}\cos^2{\sqrt{y}}$$
sepate variables
$$\displaystyle \dfrac{dy}{\sqrt{y}\, \cos^2{\sqrt{y}} }=\dfrac{1}{7}\, dx
\implies \int{\dfrac{{d}y}{\sqrt{y}\,\cos^2{\left(\sqrt{y} \right) }} }
= \int{ \dfrac{1}{7}\,{d}x}$$
ok i think u subst is next ... maybe...
$$u=\sqrt{y} \therefore du=\dfrac{{d}y}{2\,\sqrt{y}}$$
 
Last edited by a moderator:
Physics news on Phys.org
karush said:
Solve the de $$\dfrac{dy}{dx}=\dfrac{1}{7}\sqrt{y}\cos^2{\sqrt{y}}$$
sepate variables
$$\displaystyle \dfrac{dy}{\sqrt{y}\, \cos^2{\sqrt{y}} }=\dfrac{1}{7}\, dx
\implies \int{\dfrac{{d}y}{\sqrt{y}\,\cos^2{\left(\sqrt{y} \right) }} }
= \int{ \dfrac{1}{7}\,{d}x}$$
ok i think u subst is next ... maybe...
$$u=\sqrt{y} \therefore du=\dfrac{{d}y}{2\,\sqrt{y}}$$
You stopped too soon! What did you get when you did the substitution?

-Dan
 
Last edited by a moderator:
using your sub \( u = \sqrt{y} \) ...

\( \displaystyle 2 \int \dfrac{1}{\cos^2{\sqrt{y}}} \cdot \dfrac{dy}{2\sqrt{y}} = 2\int \sec^2{u} \, du \)
 
Thread 'Direction Fields and Isoclines'
I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
Back
Top